Total distance the ball travels down: 2.5 a ft. Total distance the ball travels up: 2.5 a ft. Total distance the ball travels: 5 a ft
Example of an infinite geometric series in real life: A bouncing ball is an example of an infinite geometric series in real life. The heights of each bounce of the ball can be considered a geometric sequence.
Given information: While tossing around a ball one day, you notice that when you drop the ball, the rebound height is always less than the previous height. Let us assume the initial height from which the ball is dropped be ‘a’ ft. The rebound height of the ball from the previous height is always less. Therefore, the ball bounces down by a factor of 3/5, or 0.6, of its previous height. Applying the formula of an infinite geometric series, we get;
S = a / (1 - r)
Where S = total distance the ball travels down, a = initial height from which the ball is dropped, and r = the common ratio= 0.6Substituting the values we get;
S = a / (1 - 0.6)= a / 0.4
Therefore, the total distance the ball travels down is 2.5 times the height from which the ball is initially dropped. Similarly, the total distance the ball travels up is the sum of an infinite geometric sequence with the first term of 3/5 and a common ratio of 3/5. Therefore, the total distance the ball travels up is also 2.5 times the height from which the ball is initially dropped. The total distance that the ball travels is 5 times the height from which the ball is initially dropped.
Initial height from which the ball is dropped: a = ? ft
Fraction of the previous height: r = 3/5
Initial height from which the ball is dropped: a = ? ft
Fraction of the previous height: r = 3/5
Using the values of a and r from Part 1, let us calculate the total distance the ball travels down and up.
Total distance the ball travels down:
S = a / (1 - r)
Where S = total distance the ball travels down, a = initial height from which the ball is dropped, r = the common ratio = 0.6
Substituting the values we get;
S = a / (1 - 0.6)= a / 0.4
Total distance the ball travels up:
The total distance the ball travels up is also 2.5 times the height from which the ball is initially dropped.
Total distance the ball travels:
The total distance that the ball travels is 5 times the height from which the ball is initially dropped.
Total distance the ball travels down: 2.5 a ft
Total distance the ball travels up: 2.5 a ft
Total distance the ball travels: 5 a ft
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Use the Root Test to determine whether the series converges absolutely or diverges. 4
1
+( 8
1
) 2
+( 12
1
) 3
+( 16
1
) 4
+⋯ Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer in simplified form.) A. The series diverges because rho= B. The series converges absolutely because rho= c. The Root Test is inconclusive because rho=
The Root Test states that if the limit of the nth root of the absolute value of the nth term of a series is less than 1, then the series converges absolutely.
If it is greater than 1, then the series diverges.
If it is equal to 1, then the test is inconclusive.
The formula for the nth term of the series is given by, an = (4n)/(n^2) = 4/n.
So, we need to find the limit of the nth root of the absolute value of the nth term of the series as n approaches infinity.
Let's apply the Root Test to the given series.
(4/n)^(1/n)
= 4^(1/n) / n^(1/n)
= 4^(1/n) / e
Since e > 1, 4^(1/n) / e → 0 as n approaches infinity.
Therefore, the series converges absolutely. Hence, the answer is option B.
The series converges absolutely because rho= 0.
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If p is the proposition "I want pears" and q is the proposition "I want oranges," rewrite the sentence "I do not want oranges, but I want pears" using symbols. CIDE The statement "I do not want oranges, but I want pears" can be written using symbols as
The solution to this problem is ¬q ∧ p
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Determine whether the following set equipped with the given operations is a vector space. If not vector spaces identify the vector space axioms that fail. The set of all pairs of real numbers of the form (1,x) with the operations (1,y)+(1,y ′
)=(1,y+y ′
) and k(1,y)=(1,ky) (b) Determine whether the following set equipped with the given operations is a vector space. If not vector spaces identify the vector space axioms that fail. The set of all 2×2 matrices of the form [ a
0
0
b
] with the standard matrix addition and scalar multiplication.
Determine whether the following set equipped with the given operations is a vector space. If not vector spaces identify the vector space axioms that fail. The given set is {(1, x) : x is a real number}. The given operations are:
[tex](1, y) + (1, y′) = (1, y + y′)[/tex]and k[tex](1, y) = (1, ky)[/tex]Let (a, b) and (c, d) be arbitrary elements in the set.
Now we can check whether the axioms of vector space hold for this given set or not. It can be seen that the following vector space axioms do not hold for this given set:
1. Closure under scalar multiplication:
Let k be any scalar and (a, b) be any element in the set.
Then k(1, b) = (1, kb), which is not of the form (1, x) where x is a real number. So, the set is not closed under scalar multiplication. Hence, it is not a vector space.(b) Determine whether the following set equipped with the given operations is a vector space. If not vector spaces identify the vector space axioms that fail.
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Evaluate limx→0(2x3−12x+8)
The value of limx→0 (2x³ - 12x + 8) is 8
Given function is `limx→0 (2x^3-12x+8)
To evaluate the limit of the given function, use the formula:(a³ - b³) = (a - b)(a² + ab + b²)
Using this formula, we get the function as follows : (2x³ - 12x + 8) = 2(x³ - 6x + 4)
Thus, the given function can be rewritten as `limx→0 (2x³ - 12x + 8)= limx→0 [2(x³ - 6x + 4)]
= 2 limx→0 (x³ - 6x + 4)
Now, substituting `0` for `x` in `x³ - 6x + 4`, we get= 2[0³ - 6(0) + 4]
= 2(4)
= 8
Hence, the value of `limx→0 (2x³ - 12x + 8) is 8.
Therefore, the correct option is (D) 8.
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State the conclusion based on the results of the test According to the report, the standard deviation of monthly cell phone bills was $48.12 three years ago. A researcher suspects that the standard deviation of monthly cell phone bills is different today. The null hypothesis is rejected Choose the correct answer below OA. There is not sufficient evidence to conclude that the standard deviation of monthly coll phone bills is different from its level three years ago of $48 12 B. There is sufficient evidence to conclude that the standard deviation of monthly cell phone bills is higher than its level three years ago of $48.12 OC. There is sufficient evidence to conclude that the standard deviation of monthly cell phone bills is different from its level three years ago of $48. 12.
Answer: OC. There is sufficient evidence to conclude that the standard deviation of monthly cell phone bills is different from its level three years ago of $48.12.
The given report mentions that the standard deviation of monthly cell phone bills was $48.12 three years ago, and that the researcher suspects that the standard deviation of monthly cell phone bills has changed. The null hypothesis is rejected. The conclusion that can be drawn from this is: There is sufficient evidence to conclude that the standard deviation of monthly cell phone bills is different from its level three years ago of $48.12.
The given null hypothesis says that there is no change in the standard deviation of monthly cell phone bills from three years ago. If this null hypothesis is rejected, it means that there is some evidence that the standard deviation of monthly cell phone bills has changed.
The alternative hypothesis in this case would be that the standard deviation of monthly cell phone bills is different from what it was three years ago. Since the null hypothesis is rejected, it means that there is evidence to support the alternative hypothesis.
Therefore, the conclusion that can be drawn from this is that there is sufficient evidence to conclude that the standard deviation of monthly cell phone bills is different from its level three years ago of $48.12.
Answer: OC. There is sufficient evidence to conclude that the standard deviation of monthly cell phone bills is different from its level three years ago of $48.12.
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a company sales 160 computer peice daily for $6.99
each . each of the cost $1.67 per peice plus a flat rate of
$100
The company's daily profit can be calculated by subtracting the total cost (including the cost per piece and a flat rate) from the total revenue generated from selling 160 computer pieces.
To calculate the profit of the company, we need to consider the cost and revenue. The cost per computer piece is given as $1.67, and there is a flat rate of $100. The revenue is determined by the selling price per computer piece, which is $6.99. The company sells 160 computer pieces daily, so the daily revenue is 160 * $6.99.
To calculate the cost, we multiply the cost per piece by the number of computer pieces sold daily, which is 160 * $1.67. We also need to add the flat rate cost of $100.The profit can be obtained by subtracting the total cost from the total revenue. Thus, the profit per day is (160 * $6.99) - (160 * $1.67) - $100.
In summary, the daily profit of the company can be calculated by subtracting the total cost from the total revenue, considering the selling price per computer piece, the cost per piece, and the flat rate cost.
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If we rewrite −6sin(x)−5cos(x) as A ∗
sin(x+y), what is y (let y be between −pi and pi)?
If we rewrite -6sin(x) - 5cos(x) as A×sin(x + y), then the value of y where y is between -π and π is tan⁻¹(5/6)
To find the value of y, follow these steps:
It is given that -6sin(x) - 5cos(x) can be rewritten as A×sin(x + y). So, -6sin(x) - 5cos(x)= A×sin(x + y). Using the trigonometric formula sin(x + y) = sin(x)cos(y) + cos(x)sin(y) and substituting in the equation, we get -6sin(x) - 5cos(x) = Asin(x)cos(y) + Acos(x)sin(y) ⇒-6sin(x) - 5cos(x) = (Acos(y))sin(x) + (Asin(y))cos(x).On comparing the two equations, Acos(y)= -6 and Asin(y)= -5.Dividing Asin(y)/Acos(y)= tan(y)= 5/6 ⇒y = tan⁻¹(5/6).Hence, the value of y= tan⁻¹(5/6)
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Find the surface area of the part of the plane z = 4 + 3x + 7y that lies inside the cylinder x² + y² = 1
The surface area of the part of the plane z = 4 + 3x + 7y inside the cylinder x² + y² = 1 can be found by evaluating the double integral of √59 over the region in polar coordinates.
To find the surface area of the part of the plane z = 4 + 3x + 7y that lies inside the cylinder x² + y² = 1, we can set up a double integral over the region of the cylinder.
Let's express z as a function of x and y:
z = 4 + 3x + 7y
We can rewrite the equation of the cylinder as:
x² + y² = 1
To find the surface area, we need to evaluate the double integral of the square root of the sum of the squared partial derivatives of z with respect to x and y, over the region of the cylinder.
Surface area = ∬√(1 + (∂z/∂x)² + (∂z/∂y)²) dA
∂z/∂x = 3
∂z/∂y = 7
Substituting these partial derivatives into the surface area formula, we get:
Surface area = ∬√(1 + 3² + 7²) dA
Surface area = ∬√(1 + 9 + 49) dA
Surface area = ∬√59 dA
Now, we need to determine the limits of integration for x and y over the region of the cylinder x² + y² = 1. This region corresponds to the unit circle centered at the origin in the xy-plane.
Using polar coordinates, we can parameterize the region as:
x = rcos(θ)
y = rsin(θ)
In polar coordinates, the limits of integration for r are 0 to 1, and for θ, it is 0 to 2π (a full revolution).
Now, let's convert the double integral into polar coordinates:
Surface area = ∫[0 to 2π] ∫[0 to 1] √59 * r dr dθ
Evaluating this double integral will give us the surface area of the part of the plane that lies inside the cylinder.
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The value of (01111∧10101)∨01000 is: 01101 1111 01000 10101
The value of the expression [tex](01111∧10101)∨01000[/tex]is 01101.
To calculate the value of the expression (01111∧10101)∨01000, we need to evaluate each operation separately.
First, let's perform the bitwise AND operation (∧) between the numbers 01111 and 10101:
[tex]01111∧ 10101--------- 00101\\[/tex]
The result of the bitwise AND operation is 00101.
Next, let's perform the bitwise OR operation (∨) between the result of the previous operation (00101) and the number 01000:
[tex]00101∨ 01000--------- 01101[/tex]
The result of the bitwise OR operation is 01101.
Therefore, the value of the expression ([tex]01111∧10101)∨01000[/tex] is 01101.
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The value of (01111 ∧ 10101) ∨ 01000 is 01101, which represents the decimal number 13.
The given expression is (01111 ∧ 10101) ∨ 01000. Here, ∧ represents the logical AND operator and ∨ represents the logical OR operator.
The value of the given expression is 01101 in binary, which is equivalent to 13 in decimal.
Explanation: Main part: The value of (01111 ∧ 10101) ∨ 01000 is 01101
Explanation: Let's break down the given expression into smaller parts and evaluate them one by one. First, we need to evaluate the expression (01111 ∧ 10101). To do this, we perform a bitwise AND operation between the binary numbers 01111 and 10101 as follows: 01111 (in binary)10101 (in binary)------00101 (in binary). Here, we get the binary number 00101 as the result. This represents the decimal number 5. Now, we need to evaluate the expression (5 ∨ 01000).
To do this, we perform a bitwise OR operation between the decimal number 5 and the binary number 01000 as follows: 5 (in decimal)01000 (in binary)------01101 (in binary)
Here, we get the binary number 01101 as the result. This represents the decimal number 13.Therefore, the value of the given expression is 01101 in binary, which is equivalent to 13 in decimal.
Conclusion: The value of (01111 ∧ 10101) ∨ 01000 is 01101, which represents the decimal number 13.
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Sketching Hyperbolics. On the same set of axes sketch the following graphs: y = cosh(2x); y = cosh(2x + 3); y = sech (2x + 3)
Please explain the method without calculator.
Hyperbolic functions are used to represent the relationship between the exponential function and the hyperbola. The hyperbolic sine function and the hyperbolic cosine function are among the most well-known hyperbolic functions. A graph of hyperbolics can be sketched without using a calculator.
Step 1: Sketching y=cosh(2x)
In this function, there are no phase or amplitude shifts. The graph passes through the origin, and the graph's concavity is upward. The points of inflection are at x = 0. The critical point is located at (0,1), and the function's values are greater than or equal to 1.
Step 2: Sketching y=cosh(2x+3)
When the "2x" term is replaced with "2x+3," there is a horizontal shift to the left by 3 units. This corresponds to a shift of the graph to the left by 3 units. The function's values are still greater than or equal to 1, and there are still points of inflection at x = -3/2.
Step 3: Sketching y=sech(2x+3)
This function is the reciprocal of cosh(x) and its graph is in a downward concave. When the "2x+3" term is introduced, the graph of y=sech(2x+3) shifts to the left by 3 units, similar to the other two graphs. The vertical asymptotes are located at x = -3/2 and the values of the function are less than or equal to 1.
Step 4: Final step
In the final step, combine the three graphs on the same set of axes and label them accordingly. To do this, plot the critical point (0, 1) of the first graph and mark the points of inflection. Move the graph to the left by 3 units, as shown in the second graph. Finally, plot the vertical asymptotes and place the graph below the other two, as shown in the third graph. This completes the graph of the three functions.
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Chris has invested $15,000 in a savings account that is paying interest at a 7.8% (stated rate), compounded monthly. What is the Annual Percentage Yield (APY) that Chris is earning on his investment? 4. Calculate the present value of the annuity stream of $7,000 received each quarter for six years on the first day of each quarter if your investments pay 6 percent compounded quarterly
1. The Annual Percentage Yield (APY) that Chris is earning on his investment is 8.05%.
2. The present value of the annuity stream of $7,000 received each quarter for six years on the first day of each quarter is $233,240.75.
1. To calculate Annual Percentage Yield (APY), the formula used is as follows:
APY = (1 + r/n) ^ n - 1
Where r is the stated interest rate and n is the number of compounding periods per year. Given that the investment of Chris in the savings account is $15,000 and the interest rate is 7.8% compounded monthly, the Annual Percentage Yield (APY) that Chris is earning on his investment can be calculated as follows:
Stated interest rate, r = 7.8% = 0.078
Compounding frequency, n = 12
APY = (1 + 0.078/12) ^ 12 - 1 = 0.0805 or 8.05%
Therefore, the APY is 8.05%.
2. The formula to calculate the present value of an annuity stream is given as follows:
PV = PMT [(1 - (1 + r/n)^(-n*t))]/(r/n)
Where PMT is the periodic payment (Annuity), r is the interest rate, n is the number of compounding periods per year and t is the number of years.
Given that the annuity stream is $7,000 received each quarter for six years on the first day of each quarter and investments pay 6 percent compounded quarterly, the present value of the annuity stream can be calculated as follows:
PMT = $7,000, r = 6/4 = 1.5%, n = 4, t = 6 years
PV = 7000 [(1 - (1 + 1.5%/4)^(-4*6))]/(1.5%/4) = $233,240.75
Therefore, the present value of the annuity stream is $233,240.75.
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Determine the following limit. lim x→[infinity]
15x
sin5x
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim x→[infinity]
15x
sin5x
= (Simplify your answer.) B. The limit does not exist and is neither −[infinity] nor [infinity].
The correct choice is A. lim x → ∞ 15x / sin 5x = 15/5 = 3.
Explanation:
Given limit is, lim x → ∞ 15x / sin 5xWe need to solve this limit.
To solve this limit, multiply and divide by x on the numerator.
So, lim x → ∞ (15 / 5) (5x / x) / (sin 5x / x)lim x → ∞ 3 (5 / x) / (sin 5x / x)
Here, we know that 5 / x → 0 as x → ∞.
Therefore, lim x → ∞ 3 (5 / x) / (sin 5x / x) = 3 × 0 / 1 = 0
Hence, lim x → ∞ 15x / sin 5x = 0. Therefore, A is the correct option.
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Let V and W be real vector spaces, and let T:V→W be a linear map. Show that if S⊆V is a convex set, then T(S) is a convex subset of W.
T(λv1 + (1 − λ)v2) ∈ T(S) and so, T(S) is a convex subset of W.
Let V and W be real vector spaces and let T:V → W be a linear map.
Suppose that S is a convex set in V. Then, T(S) is a convex subset of W.
Proof: To show that T(S) is convex, let w1,w2 ∈ T(S) and let λ ∈ [0,1].
There exist vectors v1,v2 ∈ S such that T(v1) = w1 and T(v2) = w2, as T is onto.
Suppose λ ∈ [0,1] and let w = λw1 + (1 − λ)w2 be a convex combination of w1 and w2.
Since T is linear, we have: λT(v1) + (1 − λ)T(v2) = T(λv1 + (1 − λ)v2)
Now λv1 + (1 − λ)v2 is a convex combination of v1 and v2 in S, and so, λv1 + (1 − λ)v2 ∈ S.
Therefore, T(λv1 + (1 − λ)v2) ∈ T(S) and so, T(S) is a convex subset of W.
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(i) Prove that S 3
=<(13),(123)>. (ii) Is {(13),(123)} a minimal generating set for S 3
? Justify your answer. (iii) Is S 6
=<(13),(1245)> ? Justify your answer.
i. () = (13)(13)
(12) = (13)(123)(13)
(23) = (123)(13)
ii. (13), (123)} is a minimal generating set for S₃.
iii. S₆ is not equal to <(13), (1245)> because it does not generate all the elements of S₆.
How do we calculate?
(i)
We will show that every element of S₃ can be generated by the elements (13) and (123), and that (13) and (123) belong to S₃.
S₃ = {(), (12), (13), (23), (123), (132)}
(13) = (123)(123) = (123)²
(123) = (13)(123) = (13)²(13)
We see that both (13) and (123) belong to S₃.
() = (13)(13)
(12) = (13)(123)(13)
(23) = (123)(13)
We can see that we can express every element of S₃ as a product of (13) and (123), and (13) and (123) belong to S₃, we can conclude that S₃ = <(13), (123)>.
(ii)
It is impossible to remove any element from {(13), (123)} and still be able to generate S₃.
Hence {(13), (123)} is a minimal generating set for S₃.
(iii) S₆ = <(13), (1245)>
Our goal here is to see if every element of S₆ can be generated by (13) and (1245), and if (13) and (1245) belong to S₆.
The elements of S₆ consist of all permutations of {1, 2, 3, 4, 5, 6}.
Since (13) swaps 1 and 3, and (1245) swaps 1 with 2 and 4 with 5, it is clear that (13) and (1245) do not generate all possible permutations of {1, 2, 3, 4, 5, 6}.
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Find the appropriate critical F-value for each of the following using the F-distribution table. a. D, 20, D₂ = 15, <=0.05 b. D₁ =9, D = 24, <= 0.05 c. D₁ =20, D₂ = 15, x=0.01 a. The critical F-value when D₁ =20, D₂ = 15, and x=0.05 is. (Round to three decimal places as needed.)
a. The critical F-value when D₁ =20, D₂ = 15, and x=0.05 is 2.845.
b. The critical F-value when D₁ =9, D₂ = 24, and x=0.05 is 2.501.
c. The critical F-value when D₁ =20, D₂ = 15, and x=0.01 is 4.384.
a. D₁ =20, D₂ = 15, and x=0.05
The critical F-value can be calculated by using the F-distribution table. Here the given values are D₁ =20, D₂ = 15, and x=0.05. The critical F-value can be calculated from the F-distribution table as 2.845. The critical F-value when D₁ =20, D₂ = 15, and x=0.05 is 2.845 (rounded to three decimal places as needed).
b. D₁ =9, D₂ = 24, and x=0.05
The critical F-value can be calculated by using the F-distribution table. Here the given values are D₁ =9, D₂ = 24, and x=0.05. The critical F-value can be calculated from the F-distribution table as 2.501. The critical F-value when D₁ =9, D₂ = 24, and x=0.05 is 2.501 (rounded to three decimal places as needed).
c. D₁ =20, D₂ = 15, and x=0.01
The critical F-value can be calculated by using the F-distribution table. Here the given values are D₁ =20, D₂ = 15, and x=0.01. The critical F-value can be calculated from the F-distribution table as 4.384. The critical F-value when D₁ =20, D₂ = 15, and x=0.01 is 4.384 (rounded to three decimal places as needed).
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Given the polynomial function f(x)=3x 4
−14x 3
+28x 2
−23x−24 a. List all of the possible rational zeros for this function b. Show how to use division to determine if x= 3
8
is a zero. Be sure to state your conclusion. c. Show how to use division to determine if x=−2 is a zero. Be sure to state your conclusion. d. Explain how you know whether or not x=−2 is a lower bound for the zeros of f(x)
a. The possible rational zeros are: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24,
±1/3, ±2/3, ±1, ±4/3, ±2, ±8/3, ±4, ±24/3.
b. x = 3/8 is not a zero of f(x).
c. x = -2 is not a zero of f(x).
d. x = -2 is not a lower bound for the zeros of f(x).
a. To find the possible rational zeros for the given polynomial function f(x) = 3x⁴ - 14x³ + 28x² - 23x - 24, we can use the Rational Root Theorem. According to the theorem, the possible rational zeros are all the ratios of the factors of the constant term (-24 in this case) divided by the factors of the leading coefficient (3 in this case).
The factors of -24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.
The factors of 3 are ±1, ±3.
Therefore, the possible rational zeros are:
±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±8/1, ±12/1, ±24/1,
±1/3, ±2/3, ±3/3, ±4/3, ±6/3, ±8/3, ±12/3, ±24/3.
Simplifying these fractions, we get:
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24,
±1/3, ±2/3, ±1, ±4/3, ±2, ±8/3, ±4, ±24/3.
b. To determine if x = 3/8 is a zero of the function, we can use synthetic division:
markdown
Copy code
3/8 | 3 -14 28 -23 -24
| 9/8 21/4 27/8 12
+________________________________
3 -5/8 49/4 4/8 -12
The remainder is -12. Since the remainder is not zero, x = 3/8 is not a zero of the function.
Conclusion: x = 3/8 is not a zero of f(x).
c. To determine if x = -2 is a zero of the function, we can again use synthetic division:
markdown
Copy code
-2 | 3 -14 28 -23 -24
| -6 40 -136 238
+________________________________
3 -20 68 -159 214
The remainder is 214. Since the remainder is not zero, x = -2 is not a zero of the function.
Conclusion: x = -2 is not a zero of f(x).
d. To determine whether x = -2 is a lower bound for the zeros of f(x), we need to analyze the behavior of the polynomial for x values less than -2. One way to do this is to evaluate the polynomial at x = -3 and x = -1 and observe the signs of the resulting values.
f(-3) = 3(-3)⁴ - 14(-3)³+ 28(-3)² - 23(-3) - 24 = 207
f(-1) = 3(-1)⁴ - 14(-1)³ + 28(-1)² - 23(-1) - 24 = -38
Since f(-3) > 0 and f(-1) < 0, the polynomial changes sign between -3 and -1. This means that there is at least one real zero of the polynomial between -3 and -1. Therefore, x = -2 is not a lower bound for the zeros of f(x).
Conclusion: x = -2 is not a lower bound for the zeros of f(x).
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A soft drink bottler is studying the resistance to internal pressure of one-liter returnable bottles. A random sample of 16 bottles is tested and the resistance to pressure is obtained. The data is shown below. Through a Goodness of Fit test to the Normal distribution by the Anderson Darling test, a practical AD value equal to 0.6726 is obtained.
For which of the following significance levels is the null hypothesis rejected?
Data
193.90
170.00
157.51
157.47
156.41
155.12
154.54
151.19
149.48
148.82
148.74
146.59
136.57
135.42
132.61
126.47
Select one:
a. 2.5%
b. 10%
c. 5%
d. 1%
Expert Ans
Answer:
The correct answer is: None of the above.
Step-by-step explanation:
To determine the significance level at which the null hypothesis is rejected in the Goodness of Fit test using the Anderson Darling test, we need to compare the obtained AD value (0.6726) with the critical values for different significance levels.
The Anderson Darling test is typically used for testing normality. In this case, the null hypothesis assumes that the resistance to pressure of the bottles follows a normal distribution.
Looking at the provided choices, we can compare the AD value with the critical values for the Anderson Darling test at different significance levels:
a. 2.5%
b. 10%
c. 5%
d. 1%
Since the AD value obtained (0.6726) does not exceed the critical value for any of the given significance levels, we cannot reject the null hypothesis at any of the provided significance levels.
Therefore, the correct answer is: None of the above.
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Find and classify all the critical points for the function f(x,y)=5x 2
−x 2
y+y 2
−8y+40 Justify your answers by showing all your work, and clearly showing all testing procedures. Hint: there are three critical points.
The function f(x, y) = 5x² - x²y + y² - 8y + 40 has one local minimum and two saddle points as its critical points.
To find the critical points of the function f(x, y) = 5x² - x²y + y² - 8y + 40, we need to find the points where the partial derivatives with respect to x and y are equal to zero.
Step 1: Find the partial derivative with respect to x (denoted as ∂f/∂x):
∂f/∂x = 10x - 2xy
Step 2: Set ∂f/∂x = 0 and solve for x:
10x - 2xy = 0
2x(5 - y) = 0
From this equation, we have two possibilities:
x = 0
5 - y = 0, which implies y = 5
Step 3: Find the partial derivative with respect to y (denoted as ∂f/∂y):
∂f/∂y = -x² + 2y - 8
Step 4: Set ∂f/∂y = 0 and solve for y:
-x² + 2y - 8 = 0
2y = x² + 8
y = (1/2)x² + 4
Step 5: Substitute the values of x from the previous steps into the equation y = (1/2)x² + 4 to find the corresponding y-values for the critical points.
For x = 0:
y = (1/2)(0)² + 4
y = 4
So, one critical point is (0, 4).
For y = 5:
y = (1/2)x² + 4
5 = (1/2)x² + 4
(1/2)x² = 1
x² = 2
x = ±√2
The two critical points are (√2, 5) and (-√2, 5).
To confirm these critical points, we need to perform the second derivative test.
Let's calculate the second partial derivatives:
Step 6: Find the second partial derivative with respect to x (denoted as ∂²f/∂x²):
∂²f/∂x² = 10 - 2y
Step 7: Find the second partial derivative with respect to y (denoted as ∂²f/∂y²):
∂²f/∂y² = 2
Step 8: Find the mixed partial derivative with respect to x and y (denoted as ∂²f/∂x∂y):
∂²f/∂x∂y = -2x
Now, substitute the critical points into the second partial derivatives:
For the critical point (0, 4):
∂²f/∂x² = 10 - 2(4) = 10 - 8 = 2
∂²f/∂y² = 2
∂²f/∂x∂y = -2(0) = 0
The determinant of the Hessian matrix (D) is calculated as follows:
D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²
D = (2)(2) - (0)²
D = 4
Since D > 0 and (∂²f/∂x²) > 0, the critical point (0, 4) corresponds to a local minimum.
For the critical points (√2, 5) and (-√2, 5):
∂²f/∂x² = 10 - 2(5) = 10 - 10 = 0
∂²f/∂y² = 2
∂²f/∂x∂y = -2(√2)
D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²
D = (0)(2) - (-2√2)²
D = -8
Since D < 0, the critical points (√2, 5) and (-√2, 5) correspond to saddle points.
Therefore:
The critical point (0, 4) is a local minimum.
The critical points (√2, 5) and (-√2, 5) are saddle points.
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There is a tall antenna on the top of a building. When a person stands 400 feet away from the building, the angle of elevation to the top of the building is 71 ∘
, and the angle of elevation to the top of the antenna is 75.3 ∘
. a. Sketch a diagram showing the building, the antenna, the angles of elevation and the person. b. Find the height of the antenna.
Height of the building as 273.2 feet and an approximate height of the antenna as 67.3 feet.
To sketch the diagram, we can draw a vertical line to represent the building. From a point 400 feet away from the building, we draw a line segment upward to represent the person's line of sight. At the top of the building, we draw a line segment extending further upward to represent the antenna. We label the angles of elevation, 71° for the top of the building and 75.3° for the top of the antenna.
We can use trigonometry to find the height of the antenna. Let's denote the height of the antenna as h. From the diagram, we have a right triangle formed by the person, the top of the building, and a horizontal line connecting the person and the base of the building. We can use the tangent function to relate the height of the building and the distance from the person to the building:
tan(71°) = height of the building / 400.
Solving for the height of the building gives us:
height of the building = 400 * tan(71°).
Similarly, we can use the tangent function to relate the height of the antenna and the distance from the person to the building:
tan(75.3°) = height of the antenna / 400.
Solving for the height of the antenna gives us:
height of the antenna = 400 * tan(75.3°).
Calculating these values gives us an approximate height of the building as 273.2 feet and an approximate height of the antenna as 67.3 feet.
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Use the Laplace transform to solve the given equation.
Use the Laplace transform to solve the given equation. y" - 8y' + 20y = tet, y(0) = 0, y'(0) = 0 y(t) = et 189²-¹006(21) + sin(21) 13 338 ecos e
The Laplace transform is used to solve the given second-order linear homogeneous differential equation. By applying the Laplace transform to the equation, we obtain the transformed equation in the s-domain.
By solving for the Laplace transform of y(t), we can find the inverse Laplace transform to obtain the solution in the time domain. Let's denote the Laplace transform of a function f(t) as F(s), where s is the complex variable in the Laplace domain. Applying the Laplace transform to the given equation, we obtain:
[tex]\[s^2Y(s) - 8sY(s) + 20Y(s) = \frac{1}{s}e^t\][/tex]
Simplifying the equation, we have:
[tex]\[Y(s)(s^2 - 8s + 20) = \frac{1}{s}e^t\][/tex]
Dividing both sides by [tex]\((s^2 - 8s + 20)\)[/tex], we can solve for Y(s):
[tex]\[Y(s) = \frac{1}{s(s^2 - 8s + 20)}e^t\][/tex]
The next step is to find the inverse Laplace transform of Y(s). We can factor the denominator of Y(s) as (s-2)(s-6). Using partial fraction decomposition, we can write Y(s) as:
[tex]\[Y(s) = \frac{A}{s} + \frac{B}{s-2} + \frac{C}{s-6}\][/tex]
Solving for A, B, and C, we find:
[tex]\[Y(s) = \frac{1}{s} - \frac{1}{s-2} + \frac{1}{s-6}\][/tex]
Now, we can use the inverse Laplace transform to find y(t). Taking the inverse Laplace transform of Y(s), we obtain:
[tex]\[y(t) = 1 - e^{2t} + e^{6t}\][/tex]
Therefore, the solution to the given differential equation with the given initial conditions is [tex]\(y(t) = 1 - e^{2t} + e^{6t}\)[/tex].
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The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 245.1 and a standard deviation of 69.5. (All units are 1000 cells/⿲L.) Using the empirical rule, find each approximate percentage below. a. What is the approximate percentage of women with platelet counts within 2 standard deviations of the mean, or between 106.1 and 384.1 ? b. What is the approximate percentage of women with platelet counts between 175.6 and 314.6? a. Approximately 68% of women in this group have platelet counts within 2 standard deviations of the mean, or between 106.1 and 384.1. (Type an integer or a decimal. Do not round.
Approximately 68% of women have platelet counts between 175.6 and 314.6. (Type an integer or a decimal. Do not round).
Given the blood platelet counts of a group of women has a bell-shaped distribution with a mean of 245.1 and a standard deviation of 69.5.
Using the empirical rule, we need to find the following percentage:
a) What is the approximate percentage of women with platelet counts within 2 standard deviations of the mean, or between 106.1 and 384.1?Empirical Rule states that the percentage of data within k standard deviations of the mean for bell-shaped distribution is approximately:(±1 standard deviation) - about 68% of the data (±2 standard deviations) - about 95% of the data (±3 standard deviations) - about 99.7% of the data.
Now, mean = 245.1 Standard Deviation = 69.5 Plugging in the values in the formula, we have; Lower Limit, L = Mean - 2 × standard deviationL = 245.1 - 2 × 69.5L = 106.1 Upper Limit, U = Mean + 2 × standard deviationU = 245.1 + 2 × 69.5U = 384.1 So, 68% of women in this group have platelet counts within 2 standard deviations of the mean, or between 106.1 and 384.1.
b) What is the approximate percentage of women with platelet counts between 175.6 and 314.6?Now, we need to convert the range to standard units.(x - mean) / standard deviationFor the lower limit, (175.6 - 245.1) / 69.5 = -0.996For the upper limit, (314.6 - 245.1) / 69.5 = 1.001
Using the Z-table, the area to the left of z = 1.001 is 0.8413 and the area to the left of z = -0.996 is 0.1587. Area between the limits is = 0.8413 - 0.1587 = 0.6826
Therefore, Approximately 68% of women have platelet counts between 175.6 and 314.6. (Type an integer or a decimal. Do not round).
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For a binomial distribution with a sample size equal to 4 and a probability of a success equal to 0.23, what is the probability that the sample will contain exactly three successes? Use the binomial formula to determine the probability. The probability that the sample will contain exactly three successes is (Round to four decimal places as needed.)
The probability that the sample will contain exactly three successes is 0.0378.
To find the probability that the sample will contain exactly three successes in a binomial distribution, we can use the given formula:
P(X = k) = (nCk) * p^k * q^(n-k)
where P(X = k) is the probability of getting k successes, n is the sample size, p is the probability of a success, q = 1-p is the probability of a failure, and (nCk) is the number of ways of choosing k successes from n trials.
In this case, k = 3. Plugging in the values, we get:
P(X = 3) = (4C3) * (0.23)³ * (0.77)^(4-3) = 4 * 0.012197 * 0.77 = 0.0378 (rounded to four decimal places)
Therefore, the probability will be 0.0378.
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The following table shows the value (in dollars) of five
external hard drives of various ages (in years). age 1 2 3 6 8
value 80 65 55 35 15
(a) Find the estimated linear regression equation.
(b) Compute the coefficient of determination r 2
a) The estimated linear regression equation is:value = 81 - 9.5*age
To find the estimated linear regression equation and compute the coefficient of determination (r^2), we can use the given data points to perform a linear regression analysis.
The linear regression equation has the form:
y = a + bx
Where:
y is the dependent variable (value in this case)
x is the independent variable (age in this case)
a is the y-intercept (constant term)
b is the slope (coefficient of x)
We can use the following formulas to calculate the slope and y-intercept:
b = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
a = (Σy - bΣx) / n
r^2, the coefficient of determination, can be calculated using the formula:
r^2 = (SSR / SST)
Where:
SSR is the sum of squared residuals (deviations of predicted values from the mean)
SST is the total sum of squares (deviations of actual values from the mean)
Using the given data points:
age: 1, 2, 3, 6, 8
value: 80, 65, 55, 35, 15
We can calculate the necessary summations:
Σx = 1 + 2 + 3 + 6 + 8 = 20
Σy = 80 + 65 + 55 + 35 + 15 = 250
Σxy = (180) + (265) + (355) + (635) + (8*15) = 705
Σx^2 = (1^2) + (2^2) + (3^2) + (6^2) + (8^2) = 110
Using these values, we can calculate the slope (b) and the y-intercept (a):
b = (5705 - 20250) / (5*110 - 20^2) = -9.5
a = (250 - (-9.5)*20) / 5 = 81
Therefore, the estimated linear regression equation is:
value = 81 - 9.5*age
b) To compute the coefficient of determination (r^2), we need to calculate SSR and SST:
SSR = Σ(y_predicted - y_mean)^2
SST = Σ(y - y_mean)^2
Using the regression equation to calculate the predicted values (y_predicted), we can calculate SSR and SST:
y_predicted = 81 - 9.5*age
Calculating SSR and SST:
SSR = (80 - 70.6)^2 + (65 - 70.6)^2 + (55 - 70.6)^2 + (35 - 51.1)^2 + (15 - 63.6)^2 = 1305.8
SST = (80 - 59)^2 + (65 - 59)^2 + (55 - 59)^2 + (35 - 59)^2 + (15 - 59)^2 = 2906
Now, we can compute r^2:
r^2 = SSR / SST = 1305.8 / 2906 ≈ 0.4494
Therefore, the coefficient of determination (r^2) is approximately 0.4494, indicating that around 44.94% of the variability in the value of the external hard drives can be explained by the linear regression model.
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each of the following random variables as either discrete or continuous: • number of students in class [Select] • distance traveled between classes (Select] • weight of students in class (Select] > • number of red marbles in a jar (Select) . time it takes to get to school (Select] • number of heads when flipping a coin three times (Select]
The random variables can be classified as follows:
Number of students in class: Discrete
Distance traveled between classes: Continuous
Weight of students in class: Continuous
Number of red marbles in a jar: Discrete
Time it takes to get to school: Continuous
Number of heads when flipping a coin three times: Discrete
In probability theory, random variables can be categorized as either discrete or continuous. A discrete random variable is one that can only take on a finite or countably infinite number of values. In this context, the number of students in a class is a discrete random variable because it can only be a whole number, such as 20, 30, or 40.
On the other hand, a continuous random variable can take on any value within a specified range or interval. The distance traveled between classes is a continuous random variable since it can be any positive real number, such as 1.5 miles, 2.3 miles, or 3.7 miles.
Similarly, the weight of students in a class is also a continuous random variable because it can take on any positive real number within a certain range, like 120 pounds, 150 pounds, or 180 pounds.
Moving on to the number of red marbles in a jar, it is a discrete random variable since it can only have integer values, such as 0, 1, 2, and so on.
The time it takes to get to school is a continuous random variable as it can take on any positive real number within a specific time frame, like 15 minutes, 20 minutes, or 25 minutes.
Lastly, the number of heads when flipping a coin three times is a discrete random variable since it can only take on a limited number of values: 0, 1, 2, or 3.
In conclusion, the classification of the given random variables is as follows: number of students in class (discrete), distance traveled between classes (continuous), weight of students in class (continuous), number of red marbles in a jar (discrete), time it takes to get to school (continuous), and number of heads when flipping a coin three times (discrete).
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A project has an initial cost of $30 million. The project is expected to generate a cash flow of $3.7 million at the end of the first year. All the subsequent cash flows will grow at a constant growth rate of 4% forever in future. If the appropriate discount rate of the project is 11%, what is the profitability index of the project?
The value of the profitability index of the project is 2.381.
We know that the growth rate is 4% and the cash flow is $3.7 million, so we can calculate the present value of all future cash flows as follows;
PV of all subsequent cash flows = 3.7 million * (1 + 0.04) / (0.11 - 0.04) = $68.1333 million
Total PV = PV of first-year cash flow + PV of all subsequent cash flows = $3.3154 million + $68.1333 million = $71.4487 million
Finally, we can calculate the profitability index as;
Profitability index = PV of future cash flows / Initial investment = $71.4487 million / $30 million = 2.381
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Solve x+3
7
= 4
x
[K13] b). Solve x 2
−x−6
24
− x+2
x−1
= 3−x
x+3
[ K
15
]
The solution to the equation x + 37 = 4x is 37/3
How to detemrine the solution to the equationfrom the question, we have the following parameters that can be used in our computation:
x + 37 = 4x
Evaluate the like terms
So, we have
3x = 37
Divide both sides by 3
x = 37/3
Hence, the solution to the equation is 37/3
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Which of the following is the integrating factor of (4xy+3y2−x)dx+x(x+2y)dy=0 that will make it an exact differential equation? a. ex2 b. x2 c. 2x d. e2x
Given differential equation is (4xy + 3y² - x)dx + x(x + 2y)dy = 0.
The integrating factor (IF) for the given differential equation is:
IF = e^(∫Pdx)
Here, P = coefficient of dx, which is (4xy + 3y² - x)/x
Thus,IF = e^(∫(4xy + 3y² - x)/x dx)
Let's find the integrating factor (IF) for each option:
(a) IF = e^(∫(4xy + 3y² - x)/x dx) = e^(∫(4y + 3y²/x - 1) dx) = e^(4xy + (3y²/x - x) dx) ≠ f(yx)
(b) IF = e^(∫(4xy + 3y² - x)/x dx) = e^(∫(4y + 3y²/x - 1) dx) = e^(4xy + (3y²/x - x) dx) ≠ f(yx)
(c) IF = e^(∫(4xy + 3y² - x)/x dx) = e^(∫(4y + 3y²/x - 1) dx) = e^(4xy + (3y²/x - x) dx) ≠ f(yx)
(d) IF = e^(∫(4xy + 3y² - x)/x dx) = e^(∫(4y + 3y²/x - 1) dx) = e^(4xy + (3y²/x - x) dx) ≠ f(yx)
Hence, there is no option that represents the integrating factor (IF) that will make the given differential equation an exact differential equation.
The answer is that there is no integrating factor for the given differential equation to make it exact.
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(T point) Consider the ellipsoid 5x 2
+y 2
+z 2
=18. The implicit form of the tangent plane to this eilipsoid at (−1,−2,−3) is The parametric form of wa int that is perpendicular to that tangent plane is L(t)= Find the point on the ornne. 2
−2y 2
at which vector n=⟨−24,16,−1⟩ is normal to the tangent plane.
The point on the curve where the vector \(\mathbf{n}\) is normal to the tangent plane is \(P = \left( -\frac{6}{5}, -\frac{3}{5}, -\frac{9}{5} \right)\).
The implicit form of the tangent plane to the ellipsoid \(5x^2 + y^2 + z^2 = 18\) at the point \((-1, -2, -3)\) is \(5x + 4y + 6z = -38\).
To find a point on the given curve \(2x^2 - 2y^2 = 0\) at which the vector \(\mathbf{n} = \langle -24, 16, -1 \rangle\) is normal to the tangent plane, we need to solve the system of equations formed by equating the parametric form of the line \(L(t)\) on the curve and the equation of the tangent plane.
Solving the equations, we find that the point on the curve where the vector \(\mathbf{n}\) is normal to the tangent plane is \(P = \left( -\frac{6}{5}, -\frac{3}{5}, -\frac{9}{5} \right)\).
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Twenty-four slips of paper are each marked with a different letter of the alphabet and placed in a basket. A slip is puiled out, is letier recorded (in the order in which the slip was drawn), and the slip is replaced. This is done 4 times. Find the probability that the word Pool is formed. Assume that each letter in the word is arso in the basket The probability is P(E)= (Use scientific notation, Round to three decimal places as needed.)
The probability of forming the word POOL is P(E) = 1/331776.
Given, twenty-four slips of paper are each marked with a different letter of the alphabet and placed in a basket. A slip is pulled out, is letter recorded (in the order in which the slip was drawn), and the slip is replaced. This is done 4 times.We have to find the probability that the word POOL is formed.
Assume that each letter in the word is also in the basket. Let's solve the problem.
There are 24 slips in a basket and a slip is pulled out 4 times with replacement.
The probability that the word POOL is formed is to be found.
Each of the letters is present on a single slip. Let the first letter be P.
There is only one slip with P on it.
Therefore, the probability of getting P is 1/24.
Similarly, there is only one slip with the letter O on it.
The probability of getting O is also 1/24.
The next letter is O again.
The probability of getting the letter O again is 1/24.
Finally, there is one slip with L on it.
The probability of getting L is 1/24.
The probability of getting POOL is
P(E) = (1/24) × (1/24) × (1/24) × (1/24)
= [tex](1/24)^4.[/tex]
The probability of getting the word POOL is 1/331776.
Therefore, the probability that the word POOL is formed is P(E) = 1/331776.
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The quadratic function, h(t)=−16t 2
+32t+64 models the height, h (in feet) of an object after t seconds when the object is thrown from ground. How long will it take for the object to return to the ground? 1− 5
s and 1+ 5
s 1+ 5
s 1s 1− 5
s
From the quadratic equation, we determine object takes [tex]\(1 + \sqrt{5}\)[/tex] seconds to return to the ground.
To determine when the object will return to the ground, we need to find the value of t when the height h(t) is equal to zero.
The quadratic function given is h(t) = -16t² + 32t + 64. We set h(t) to zero and solve for t:
0 = -16t² + 32t + 64
Dividing the entire equation by -16 to simplify, we have:
0 = t² - 2t - 4
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, the quadratic equation does not factor easily, so we will use the quadratic formula:
[tex]\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
For our equation t² - 2t - 4 = 0, we have a = 1, b = -2, and c = -4. Substituting these values into the quadratic formula, we get:
[tex]\[t = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)}\][/tex]
[tex]\[t = \frac{2 \pm \sqrt{4 + 16}}{2}\][/tex]
[tex]\[t = \frac{2 \pm \sqrt{20}}{2}\][/tex]
[tex]\[t = \frac{2 \pm 2\sqrt{5}}{2}\][/tex]
Simplifying further, we have:
[tex]\[t = 1 \pm \sqrt{5}\][/tex]
Since time cannot be negative, we can disregard the negative value and take the positive value:
[tex]\[t = 1 + \sqrt{5}\][/tex]
Therefore, it will take [tex]\(1 + \sqrt{5}\)[/tex] seconds for the object to return to the ground.
Quadratic functions have various applications in different fields, including physics, engineering, economics, and computer science. They can be used to model various real-world phenomena such as projectile motion, optimization problems, and revenue/profit functions.
Solving quadratic equations, which involve setting a quadratic function equal to zero, can be done using different methods such as factoring, completing the square, or using the quadratic formula.
These methods help us find the roots or solutions of the equation, which correspond to the x-values where the graph of the quadratic function intersects the x-axis.
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