Step-by-step explanation:
A quadratic equation is a equation with a degree of 2. Below are the 5 ways to solve a quadratic equation.
Factoring - First, make sure that one side of the equation is 0. Turn the equation into a product of two binomials and set each one to 0. Solve both to get both solutions.Completing the Square - Add to both sides such that one side becomes a square of a binomial. Square root both sides and solve with algebra.Square Roots - If one side of the equation is a perfect square, square root both sides and solve normally.Graphing - Make one side of the equation 0 and graph. The x-intercepts are the solutions.Quadratic Formula - First, make the equation follow the form [tex]ax^2+bx+c=0[/tex]. Then, use the formula [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex] to get the solutions.Let σ(n) be the sum of all positive divisors of the integer n and let p be any prime number.
Show that σ(n) < 2n holds true for all n of the form n = p²
The statement that "σ(n) < 2n holds true for all n of the form n = p²" has been proved.
Let p be any prime number, and let σ(n) be the sum of all positive divisors of the integer n.
As p is a prime number, and 2 is the smallest prime number, so, p[tex]\geq[/tex]2
So, the positive divisors of the integer n are: 1,p,p².
As σ(n) represents the sum of all positive divisors of the integer n.
σ(n)=1+p+p²
In order to prove that σ(n) < 2n,for all n of the form n = p².
1+p+p²<2p²
p²-p-1>0
It is know that, p[tex]\geq[/tex]2.
So, p²-p-1[tex]\geq[/tex]1
Thus, σ(n) < 2n holds true for all n of the form n = p².
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plssssssssssssss help
Answer:
b. 30 m²
Step-by-step explanation:
find the area of the rectangle and triangle first
area of rectangle = length x height
aR = 10 x 6
aR = 60
area of triangle = base x height x 1/2
aT = 10 x 3 x 1/2
aT = 30
now, to find the area of the shaded area subtract the triangle from the rectangle
area = rectangle - triangle
a = 60 - 30
a = 30
2. One day at Coffee Town a customer asks for exactly 300 ml of coffee. To this 300 ml of
coffee,
the customer wants pure cream added until the cream represents 20% of the total volume of
the drink (cream and coffee. Not just 20% of the 300 ml). How much pure cream should the
barista add?
Answer:60 ml
Step-by-step explanation:20 % of 300 = 60 and snce the rest of the the l are coffee thats 20%
help please it would make my day pleaes help me
Answer:
I think it would be D
HELP PLS!!!!!!!!!!!!
Answer:(3+3)(1+3) = 24
Step-by-step explanation: There is a 1 and 3 3's. We know that added up, the sum is not 24. So it can't be all addition. But, 3 is a factor of 6 which is a factor of 24. There are 2 3's, so we add them up to get 6. Then, we have a 1 and a 3 left. we add these up to get 4. 6x4= 24.
The graph of f(x) = x2 is translated to form g(x) = (x – 5)2 + 1. On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4). Which graph represents g(x)? On a coordinate plane, a parabola opens up. It goes through (2, 10), has a vertex at (5, 1), and goes through (8, 10). On a coordinate plane, a parabola opens up. It goes through (2, 8), has a vertex at (5, negative 11), and goes through (8, 8). On a coordinate plane, a parabola opens up. It goes through (negative 8, 10), has a vertex at (negative 5, 1), and goes through (negative 2, 10).
The graph is shown in the attached image.
A two-digit number has two less units than tens. The difference
between twice the number and the number reversed is 93. Find
the number.
The required two-digit number is 75.
What is arithmetic?In mathematics, it deals with numbers of operations according to the statements.
let the number in tenth place be x.
And has two fewer units than tens.
= x - 2
The two-digit number can be given as
= 10x + (x - 2)
= (11x - 2)
Reversing the digits
= (x - 2)10 + x
= 11x -20
The difference between twice the number and the number reversed is 93.
[tex]2(11x - 2) - (11x - 20) = 93\\22x-4-11x+20 = 93\\11x+16 = 93\\11x = 93-16\\11x = 77\\x = 7[/tex]
Now the required two-digit number is,
= 11x -2
= 11*7 - 2
= 77 -2
= 75
Thus the required two-digit number is 75.
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The table gives a partial set of values of a polynomial h(x), which has a leading coefficient of 1. x –2 0 1 2 3 h(x) 0 –12 0 8 0 If every x-intercept of h(x) is shown in the table and has a multiplicity of one, what is the equation of the polynomial function?
Using the Factor Theorem, the equation of h(x) is given as follows:
h(x) = -2(x³ - 2x² - 5x + 6)
What is the Factor Theorem?The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.
Looking at the table, considering the values of x when h(x) = 0, the roots of h(x) are given as follows:
[tex]x_1 = -2, x_2 = 1, x_3 = 3[/tex]
Then the rule is:
h(x) = a(x + 2)(x - 1)(x - 3)
h(x) = a(x² + x - 2)(x - 3)
h(x) = a(x³ - 2x² - 5x + 6)
The h-intercept is of -12, as when x = 0, h = -12, hence this is used to find a as follows:
6a = -12
a = -12/6
a = -2.
Hence the function is given by:
h(x) = -2(x³ - 2x² - 5x + 6)
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Please help!!!!!!!!!!!!!!!!
Answer: Option (3)
Step-by-step explanation:
[tex]\frac{1}{6a^2}-\frac{5b}{3a^3}+\frac{b^2}{4a}\\\\=\frac{2a}{12a^3}-\frac{20b}{12a^3}+\frac{3a^2 b^2}{12a^3}\\\\=\frac{3a^2 +b2 +2a-20b}{12a^3}[/tex]
Which features are present
in this polar graph?
The features that exist present in this polar graph is that D. Symmetry about the polar axis θ = 0.
What is polar graph?A polar curve exists as a form constructed utilizing the polar coordinate system. Polar curves exists determined by points that exist at a variable distance from the origin (the pole) depending on the angle calculated off the positive x-axis.
The polar graph can be said to contain a symmetry about the polar axis which indicates that theta or θ exists equivalent to 0.
This exists because the graph is symmetric about the x-axis. This indicates that all the lines of symmetry can be seen intersecting the graph at x = 0. This demonstrates that there exists indeed symmetry in the polar graph.
Therefore, the correct answer is option D. symmetry about the plane θ = 0
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A photograph is 3 in. longer than it is wide. When a 2-in. border is placed around the photograph, the total area of the photograph and the border is 108 in^2. Find the dimensions of the photograph.
width = 7
length = 10
Step-by-step explanation:
solve for the one letter that gets talked about the most
which is w or width
width = w
length = 3 + w
2 inch border means
width = w + 2
length = 3 + w + 2
area = length times width
area = ( 3 + w + 2 ) times (w + 2 )
108 = ( 3 + w + 2 ) times (w + 2 )
108 = ( w + 5 ) times (w + 2 )
( w + 5 ) times (w + 2 ) = 108
w^2 + 7w +10 = 108
w^2 + 7w - 98 = 0
going to make w = x
x^2+7x-98=0
(x-7)(x+14)
x = 7
x = -14
w=7
w= -14
cant have a negative measurement so w=7 is used
width = w
length = 3 + w
width = 7
length = 3 + 7 = 10
Triangle ABC is a non-right triangle with Angle C = 117, a = 12, and b = 18. Find Angle B to the nearest tenth.
Using the law of cosines and sines, the measure of angle B is: 38.4°.
What is the Law of Cosines and Sines?Law of cosines is: c = √[a² + b² ﹣ 2ab(cos C)]
Law of sines is: sin A/a = sin B/b = sin C/c
Use the law of cosines to find c:
c = √[12² + 18² ﹣ 2(12)(18)(cos 117)]
c ≈ 25.8
Use the law of sines to find angle B:
sin B/b = sin C/c
sin B/18 = sin 117/25.8
sin B = (sin 117 × 18)/25.8
sin B = 0.6216
B = sin^(-1)(0.6216)
B = 38.4°
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what fraction of four weeks is 8 days
Answer:
For weeks are 28 days, so the fraction is 28/8= 3,5 and the fraction is 7/2
A _[blank 1]_ is a number whose only factors are 1 and itself. If a number has other factors besides 1 and itself, it is called a _[blank 2]_. You can use divisibility rules or _[blank 3]_ to help you determine whether a number is prime or composite.
Match each blank with the option that correctly fills in that blank.
1. the sieve of Eratosthenes.
2.whole number.
3. composite number.
4.counting number.
5. the standard algorithm
6. prime number.
Numbers whose only factors are 1 and itself is known as prime numbers , numbers whose factors besides 1 and itself are composite numbers, we can use the sieve of Eratosthenes to determine whether the number is prime or composite.
Given a paragraph in which there are blanks:
A _____ is a number whose only factors are 1 and itself. If a number has factors besides 1 and itself, it is called a _____. You can use divisibility rules or _______ to help you determine whether a number is prime or composite.
We are required to fill the blank with appropriate options.
We have to fill "prime numbers" in the first blank.
We have to fill "composite numbers" in the second blank.
We have to fill "the sieve of Eratosthenes" in the third blank.
Hence numbers whose only factors are 1 and itself is known as prime numbers , numbers whose factors besides 1 and itself are composite numbers, we can use the sieve of Eratosthenes to determine whether the number is prime or composite.
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A man weighs 51 kilograms. His brother weighs 10 percent more. How heavy is the brother? 5.1 kilograms 51.6 kilograms 55.6 kilograms 56.1 kilograms
Answer: 56.1 kilograms
Step-by-step explanation:
Solution 1: Multiply 51 by 10% and we'll get 5.1. Then, add 51 and 5.1 and we'll get 56.1
Solution 2: Multiply 51 by 110% and we'll get 56.1
(we can multiply 51 by 1 + 10% because multiplying 51 by 1 is the same thing as multiplying 51 by 100%. So you can just use this slicker formula for solving problems like this)
A company reports cost of goods manufactured of $918,700 and cost of goods sold of $955,448. Compute the average manufacturing cost per unit assuming 18,374 units were produced.
If the cost of goods manufactured of $918,700 and cost of goods sold is $955,448. The average manufacturing cost per unit assuming 18,374 units were produced is $102 per unit.
Average manufacturing cost per unitUsing this formula to determine the average manufacturing cost per unit
Average manufacturing cost per unit= Total cost/Number of units produced
Where:
Total cost=$918,700+$955,448=$1,874,148
Number of units produced=18,374 units
Let plug in the formula
Average manufacturing cost per unit=$918,700+$955,448/18,374
Average manufacturing cost per unit=$1,874,148/18,374
Average manufacturing cost per unit=$102 per unit
Therefore the average manufacturing cost per unit assuming 18,374 units were produced is $102 per unit.
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what is the midpoint of a segment with endpoints (-3, 2) and (7, -5)?
Answer:
[tex](2,-1.5)[/tex]
Step-by-step explanation:
The midpoint formula goes as follows:
[tex](x,y)=(\frac{x1+x2}{2} , \frac{y1+y2}{2})[/tex]
Lets plug in our values, let x1 be -3, x2 be 7, y1 be 2, y2 be -5.
[tex](\frac{-3+7}{2} , \frac{2+-5}{2})[/tex]
Solve numerators
[tex](\frac{4}{2} , \frac{-3}{2})[/tex]
Solve
[tex](2 , -1.5)[/tex]
Therefore the midpoint must be at [tex](2,-1.5)[/tex]
A stamp collection consisting of 23 stamps includes 4¢ stamps and 9¢ stamps. The total value of the stamps is $1.27. Find the number of each type of stamp in the collection.
A simultaneous equation is a set of equations that has to be solved in relation to each other at the same time. Thus the required number of stamps are:
4¢ stamps = 16
5¢ stamps = 7
A simultaneous equation is a set of equations that has to be solved in relation to each other at the same time. This process is required so as to determine the values of two unknowns e.g x and y.
From the given question, let the number of 4¢ stamps be represented by n, and that of the 9¢ stamp be represented by m.
So that,
n + m = 23 ............ 1
But 100¢ = $1, so that;
4¢ = x
x = [tex]\frac{4}{100}[/tex]
= $0.04
also,
9¢ = x
x = [tex]\frac{9}{100}[/tex]
= $0.09
Thus, we have;
0.04n + 0.09m = 1.27 ......... 2
From equation 1, make n the subject of the formula, such that;
n = 23 - m ........... 3
Substitute equation 3 into equation 2
0.04(23 - m) + 0.09m = 1.27
0.92 - 0.04m + 0.09m = 1.27
collect like terms to have;
0.05m = 1.27 - 0.92
= 0.35
m = [tex]\frac{0.35}{0.05}[/tex]
m = 7
Now substitute the value of m into equation 3
n = 23 - m ........... 3
= 23 - 7
n = 16
Therefore the number of 4¢ stamps is 16, while that of the 5¢ stamps is 7.
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Which of the following are true statements about a 30 69 90 triangle? Check all that apply
By the 30 - 60 - 90 theorem the right options of the multiple choice question is:
A. The hypotenuse is twice as long as the shorter leg.
D. The longer leg is √3 times as long as the shorter leg.
What are the properties of a 30 - 60 - 90 right triangles
In this problem we must look for the right choices in a multiple choice question. According to the Euclidean geometry, 30 - 60 - 90 have the following properties:
The length of the hypotenuse is √3 / 2 times as long as the longer leg.The length of the hypotenuse is twice as long as the shorter leg.The length of the longer leg is √3 times as long as the shorter leg.Thus, the right options of the multiple choice question is:
A. The hypotenuse is twice as long as the shorter leg.
D. The longer leg is √3 times as long as the shorter leg.
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1) The following scatterplot shows the percentage of the vote a candidate received in the 2016 senatorial elections
according to the voter's income level based on an exit poll of voters conducted by a news agency. The income
levels 1-8 correspond to the following income classes:
1 = Under $15,000; 2 = $15-30,000; 3 = $30-50,000; 4 = $50-75,000; 5 = $75-100,000;
6 = $100-150,000; 7 = $150-200,000; 8 = $200,000 or more.
Use the election scatterplot to determine whether there is a correlation between percentage of vote and income
level at the 0.01 significance level with a null hypothesis of ρs = 0.
A) The test statistic is between the critical values, so we fail to reject the null hypothesis. There is no
evidence to support a claim of correlation between percentage of vote and income level.
B) The test statistic is not between the critical values, so we fail to reject the null hypothesis. There is no
evidence to support a claim of correlation between percentage of vote and income level.
C) The test statistic is between the critical values, so we reject the null hypothesis. There is sufficient
evidence to support a claim of correlation between percentage of vote and income level.
D) The test statistic is not between the critical values, so we reject the null hypothesis. There is sufficient
evidence to support a claim of correlation between percentage of vote and income level
The answer to the question is B. The test statistic is not between the critical values, so we reject the null hypothesis. There is sufficient evidence to support a claim of correlation between percentage of vote and income level.
What is the scatter plot?This is the plot that shows the relationship between two different variables along a straight line. All of the points that are known to have a relationship between these variables would fall under this line. Other parts that fall outside the line are regarded as the outliers in the plot.
In this particular question, the outlier is seen to be outside of the critical values so we have to conclude that the solution is B. We fail to reject the null hypothesis.
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Suppose that a and b are integers with a < b How many numbers are in the list a, a+1, a+2.... b?
So I thought about doing a-(a-1) to get the first number to one so the list becomes 1,2,3 but i soon realized that does not work
The count of numbers in the list a, a+1, a+2.... b is b - a + 1
How to determine the count of numbers in the list a, a+1, a+2.... b?The list of numbers is given as:
a, a+1, a+2.... b
From the above list, we can see that the numbers are consecutive numbers.
This means that, the count of numbers in the list is
Count = Highest - Least + 1
Where
Highest = b
Least = a
Substitute the known values in the above equation
Count = b - a + 1
Hence, the count of numbers in the list a, a+1, a+2.... b is b - a + 1
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8x+4y=7
6x-8y=41
using ELIMINATION METHOD
Answer:
([tex]\frac{5}{2}[/tex] , - [tex]\frac{13}{4}[/tex] )
Step-by-step explanation:
8x + 4y = 7 → (1)
6x - 8y = 41 → (2)
multiplying (1) by 2 and adding to (2) will eliminate y
16x + 8y = 14 → (3)
add (2) and (3) term by term to eliminate y
22x + 0 = 55
22x = 55 ( divide both sides by 22 )
x = [tex]\frac{55}{22}[/tex] = [tex]\frac{5}{2}[/tex]
substitute this value into either of the 2 equations and solve for y
substituting into (1)
8([tex]\frac{5}{2}[/tex] ) + 4y = 7
20 + 4y = 7 ( subtract 20 from both sides )
4y = - 13 ( divide both sides by 4 )
y = - [tex]\frac{13}{4}[/tex]
solution is ( [tex]\frac{5}{2}[/tex] , - [tex]\frac{13}{4}[/tex] )
Determine the real life solutions of 9x^2+6x+1=0
Answer:
1 (double root)
Step-by-step explanation:
Given quadratic equation:
[tex]9x^2+6x+1=0[/tex]
To find the solutions to the given quadratic equation, factor the equation.
To factor a quadratic in the form [tex]ax^2+bx+c[/tex], find two numbers that multiply to ac and sum to b.
[tex]\implies ac=9 \cdot 1=9[/tex]
[tex]\implies b=6[/tex]
Therefore, the two numbers are: 3 and 3.
Rewrite the middle term as the sum of these two numbers:
[tex]\implies 9x^2+3x+3x+1=0[/tex]
Factor the first two terms and the last two terms separately:
[tex]\implies 3x(3x+1)+1(3x+1)=0[/tex]
Factor out the common term (3x + 1):
[tex]\implies (3x+1)(3x+1)=0[/tex]
Therefore:
[tex]\implies(3x+1)^2=0[/tex]
This means that the curve has one root with multiplicity 2. So the curve touches the x-axis and bounces off the axis at one point. (See attached graph).
Apply the zero-product property:
[tex]\implies 3x+1=0 \implies x=-\dfrac{1}{3}[/tex]
Therefore there is one real solution of the given quadratic equation.
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A glass vase weighs 0.17 lb. How much does a similarly shaped vase of the same glass weigh if each dimension is 6 times as large?
Answer:
7,647.4?
Step-by-step explanation:
Questions are in the picture
The value of x which S(x) is a global minimum is x = 1/2
Find a formula for S(x)From the question, we have:
x is a positive numberthe sum of its reciprocal and four times the product of x is the smallest possibleThis means that:
S(x) = 1/x + 4x^2
The domain of xFrom the question, we understand that x is a positive number.
This means that the domain of x is x > 0
As a notation, we have (0, ∞)
The value of x which S(x) is a global minimumRecall that:
S(x) = 1/x + 4x^2
Differentiate the function
S'(x) = -1/x^2 + 8x
Set to 0
-1/x^2 + 8x = 0
Multiply through by x^2
-1 + 8x^3 = 0
Add 1 to both sides
8x^3 = 1
Divide by 8
x^3 = 1/8
Take the cube root of both sides
x = 1/2
To prove the point is a global minimum, we have:
S'(x) = -1/x^2 + 8x
Determine the second derivative
S''(x) = 2/x^3 + 8
Set x = 1/2
S''(x) = 2/(1/2)^3 + 8
Evaluate the exponent
S''(x) = 2/1/8 + 8
Evaluate the quotient
S''(x) = 16 + 8
Evaluate the sum
S''(x) = 24
Because S'' is positive, then the single critical point is a global minimum
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60% of people who purchase sports cars are men. If
10 sports car owners are randomly selected, find
the probability that exactly 7 are men.
The probability that exactly 7 are men out of 10 sports car owners who are randomly selected is 0.215 given that 60% of people who purchase sports cars are men. This can be obtained by using binomial distribution formula.
Calculate the required probability:This question can be solved using binomial distribution formula.
The formula for binomial distribution is the following,
P(X) = ⁿCₓ pˣ qⁿ⁻ˣ
where,
n = number of trials(or the number being sampled)
x = number of success desired
p = probability of getting a success in one trial
q = 1 - p = probability of getting a failure in one trial
Here in the question it is given that,
⇒ 60% of people who purchase sports cars are men
This statements clearly means that probability of men purchase sports cars is 60%.
⇒ P(men purchasers) = p = 60% = 0.6
From this we can find the probability of women who purchase sports cars,
⇒ P(women purchasers) = q = 1 - p = 1 - 0.6 = 0.4
So we can find the probability that exactly 7 are men out of 10 sports car owners who are randomly selected
It is a binomial case with n = 10
By using the formula for binomial distribution we get,
P(X = 7) = ¹⁰C₇ × 0.6⁷ × 0.4³
P(X = 7) = 120 × 0.0279936 × 0.064
P(X = 7) = 0.21499
P(X = 7) = 0.215
Hence the probability that exactly 7 are men out of 10 sports car owners who are randomly selected is 0.215 given that 60% of people who purchase sports cars are men.
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For the straight line y =1/2x-4 what are the slope and y intercept
Answer:
Step-by-step explanation:
slope-- 1/2
y intercept-- -4
HELP PLEASE
What is the area of the kite below?
Answer:
Area = 33 inches squared
Step-by-step explanation:
The formula for the area of a kite is:
Area = ½ × (d)1 × (d)2
To find the first diagonal (d1):
d1 = 7 in + 4 in = 11 in
To find the second diagonal (d2):
The triangles on the right of the kite must be 3 4 5 triangle since it is a right triangle with a hypotenuse of 5 and a side of 4, we know half of a diagonal would be 3 inches.
d2 = 3 in + 3 in = 6 in
Now we can plug the diagonals into the area formula.
Area = ½ × 11 × 6 = 33 inches squared
Answer: A = 33 in²
Step-by-step explanation:
Given information:
Longer side = 7 in
Shorter side = 4 in
Shorter Hypotenuse = 5 in
Given formula:
A = (1/2) × (d₁) × (d₂)
A = aread₁ = The longer line (horizontal)d₂ = The shorter line (vertical)Please refer to the attachment below for a graphical understanding
Determine the length of the shorter side (vertical) through the Pythagorean theorem
a² + b² = c² (Pythagorean theorem)
(4)² + b² = (5)² (Substitute values into the equation)
16 + b² = 25
b² = 25 - 16
b² = 9
b = 3 or b = -3 (rej)
Substitute values into the given formula
A = (1/2) × d₁ × d₂
A = (1/2) × (7 + 4) × (3 + 3)
Simplify values in the parenthesis
A = (1/2) × 11 × 6
Simplify by multiplication
A = (11/2) × 6
[tex]\Large\boxed{A=33}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
A one to one function is given. Write an equation for the inverse function
M(x)=4x^3-3
Step-by-step explanation:
M(x) = 4x^3 - 3
let M(x) = Y
Y = 4x^3 - 3...swap places of x & y.
X = 4y^3 - 3
look for x in terms of y.
4y^3 = x + 3
Y^3 = x + 3 / 4...put both sides under cube root.
y = ( x + 3 / 4 )^1/3
ASAP help me with this
Answer:
2160°
Step-by-step explanation:
[tex]180(14-2)=2160^{\circ}[/tex]
