Question 6(Multiple Choice Worth 1 points)
(08.03 MC)

A portion of the quadratic formula proof is shown. Fill in the missing reason.


Statements Reasons
ax2 + bx + c = 0 Given
ax2 + bx = −c Subtract c from both sides of the equation
x squared plus b over a times x equals negative c over a Divide both sides of the equation by a
x squared plus b over a times x plus the quantity b over 2 times a squared equals negative c over a plus the quantity b over 2 times a squared Complete the square and add the quantity b over 2 times a squared to both sides
x squared plus b over a times x plus the quantity b over 2 times a squared equals negative c over a plus b squared over 4 times a squared Square the quantity b over 2 times a on the right side of the equation
x squared plus b over a times x plus the quantity b over 2 times a end quantity squared equals negative 4 times a times c over 4 times a squared plus b squared over 4 times a squared Find a common denominator on the right side of the equation
x squared plus b over a times x plus the quantity b over 2 times a squared equals b squared minus 4 times a times c all over 4 times a squared Add the fractions together on the right side of the equation
quantity x plus b over 2 times a end quantity squared equals b squared minus 4 times a times c all over 4 times a squared ?
Rewrite the perfect square trinomial as a binomial squared on the left side of the equation
Take the square root of both sides of the equation
Multiply both sides of the equation by 2
Square the left side of the equation

The equations that are true for x = -2 and x = 2 are:

x² – 4 = 0

4x² = 16

A quadratic equation is a type of polynomial equation of degree 2, which means that the highest exponent of the variable is 2. It can be written in the standard form:

ax² + bx + c = 0

where a, b, and c are constants and x is the variable. The term "quadratic" comes from the Latin word "quadratus," meaning "square."

Quadratic equations can have one or two real solutions, depending on the values of a, b, and c. The solutions can be found using various methods, such as factoring, completing the square, or using the quadratic formula:

x = (-b ± √(b²-4ac)) / 2a

The equations that are true for x = -2 and x = 2 are:

x² – 4 = 0

4x² = 16

The other three equations are not true for x = -2 and x = 2:

a) x² = -4 has no real solutions because the square of any real number is always non-negative, so x² can never be equal to a negative number.

b) 3x² + 12 = 0 simplifies to x² + 4 = 0, which has no real solutions because x² is always non-negative and can never be equal to -4.

2(x – 2)² = 0 simplifies to (x - 2)² = 0, which has only one solution at x = 2.

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Check the picture below.

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=\sqrt{a^2 + o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{x}\\ a=\stackrel{adjacent}{3}\\ o=\stackrel{opposite}{4} \end{cases} \\\\\\ x=\sqrt{ 3^2 + 4^2}\implies x=\sqrt{ 9 + 16 } \implies x=\sqrt{ 25 }\implies x=5[/tex]

Mary's estimate of the population proportion would be more precise than Diego's.

Part A: The sample proportion (p) for Diego's data can be found by dividing the number of students who watch television shows on a computer (65) by the total number of students in the sample (100).

p = 65/100 = 0.65

Part B: To determine if the sample is large enough to calculate a confidence interval, we can use the rule of thumb that both np and n(1-p) should be greater than or equal to 10.

np = 100 * 0.65 = 65

n(1-p) = 100 * (1 - 0.65) = 100 * 0.35 = 35

Since both values are greater than or equal to 10, Diego's sample is large enough to calculate a confidence interval for the population proportion.

Part C: The standard error (SE) can be found using the formula:

SE = sqrt[p(1-p)/n]

SE = sqrt[0.65(1-0.65)/100]

SE = sqrt[0.65 * 0.35 / 100]

SE = sqrt[0.2275/100]

SE = sqrt[0.002275]

SE ≈ 0.048 (rounded to the nearest hundredth)

Part D: Using the empirical rule (68-95-99.7 rule), we can calculate the confidence intervals for Diego's sample proportion:

68% confidence interval = p ± 1 * SE = 0.65 ± 1 * 0.048 = (0.65 - 0.048, 0.65 + 0.048) = (0.602, 0.698)

95% confidence interval = p ± 2 * SE = 0.65 ± 2 * 0.048 = (0.65 - 0.096, 0.65 + 0.096) = (0.554, 0.746)

99.7% confidence interval = p ± 3 * SE = 0.65 ± 3 * 0.048 = (0.65 - 0.144, 0.65 + 0.144) = (0.506, 0.794)

Part E: Since Mary took a larger sample (200 students) compared to Diego (100 students), we would expect her 95% confidence interval to be narrower than Diego's. A larger sample size usually leads to a smaller standard error, which in turn results in a narrower confidence interval.

This means that Mary's estimate of the population proportion would be more precise than Diego's.

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Answer: D

Step-by-step explanation:

The given expression is 2w3 + 7w2 - 4w.

The power of the expression is the highest exponent of the variable w, which is 3. However, this is not one of the options given.

A factor of the expression is a term or expression that divides the given expression completely. -4w is not a factor of the given expression.

The coefficient of a term is the numerical factor that is multiplied by the variable. The coefficient of the first term 2w3 is 2, and the coefficient of the second term 7w2 is 7. The coefficient of the last term -4w is -4. Therefore, the statement "The coefficient of the expression is w" is false.

The given expression has three terms: 2w3, 7w2, and -4w. Therefore, the statement "The expression has 3 terms" is true.

The unit rate for points per game is determined by dividing the total points scored by the number of games played. In this case, 62 points  3 games = 20.67 points per game (rounded to two decimal places).

To determine the unit rate for points per game, we need to divide the total number of points scored by the number of games played. In this case, Pedro scored 62 points over the last 3 basketball games.

The unit rate for points per game is found by dividing the total points scored by the number of games played. In this case, Pedro scored 62 points over 3 basketball games, so the unit rate for points per game is:

62 points ÷ 3 games = 20.67 points per game (rounded to two decimal places)

So, none of the given options is correct. The correct answer is 62 over 3 points per game, which is equivalent to 20.67 points per game when rounded to two decimal places.

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Answer:

62.01 i took the test

Step-by-step explanation:

The scale factor in the example given, which employs 10 blocks, is 62.5.

We must first make an educated guess as to how many blocks make up your model.

Suppose you utilized 10 blocks.

Since we know that each block is 2 feet tall, the model's height is:

10*2ft = 20ft

Given that the Empire State Building is 1,250 feet tall, we would like to adjust the scale from 20 feet to 1250 feet.

We divide the larger value by the smaller value to obtain that:

k = 1250ft/20ft = 62.5

This indicates that each foot on the model corresponds to 62.5 feet on the actual structure.

Therefore, the scale factor in the example given, which employs 10 blocks, is 62.5.

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Answer:

x>7/3 and x≤ -7/3

Step-by-step explanation:

Solve for x in the inequality

−3x+8<15 is x>7/3 the to find a second solution set, flip the inequality and the switch the sign x≤ -7/3

Answer:

Step-by-step explanation:

−3x+8<15

2x> -7

divide both side by 3

x→ [tex]-\frac{7}{3}[/tex]

Answer: The area of Ryan's chessboard is given as 64x^6y^8 square units.

We know that the area of a square is given by A = s^2, where A is the area and s is the side length.

To find the side length of the square, we can use the formula for the side length of a square in terms of its area, which is s = √A.

Substituting A = 64x^6y^8, we get:

s = √(64x^6y^8)

Using the properties of square roots, we can simplify this expression as follows:

s = √(2^6 * (x^2)^3 * (y^2)^4)

s = √(2^6) * √((x^2)^3) * √((y^2)^4)

s = 2^3 * x^3 * y^4

Therefore, the expression for the side length of Ryan's chessboard is 2^3 * x^3 * y^4.

Step-by-step explanation:

Answer:

No, (-2, 3) is not a solution

Step-by-step explanation:

Substitute x with -2 and y with -3.

x = -2

-2 = -2   true

2x + 3y = 6

2(-2) + 3(3) = 6

-4 + 9 = 6

5 = 6    false

The given expression rs+10s−5r−50 can be factorized by grouping as (r+10)(s-5)

Factorization is the process of breaking down an expression into simpler factors. In mathematics, factorization is often used to simplify expressions, solve equations, and find common factors.Factorization is an important tool in algebra and is used in many areas of mathematics, including number theory, geometry, and calculus.

The given terms can be reqrouped into two proportional parts as ; (rs+10s) +(-5r - 50), the we can Factor the expression as ;

s(r+10) +(-5r - 50)

s(r+10) -5(r + 10)

(r+10)(s-5)

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The statements provided about fractions and percentage of yogurt cups and the strawberry flavor are, respectively, true, false, true, false.

In a case of 30 yogurt cups, 6 out of every 10 are strawberry. Therefore, there are a total of 18 strawberry yogurt cups in the case.

To find the fraction of yogurt cups that are strawberry, you divide the total number of strawberries (18) by the total number of cups (30), which gives 18/30.

The total number of strawberry yogurt cups is 18, based on the fact that there are 30 yogurt cups in the case.

60% of the yogurt cups are strawberry, which is the result of multiplying the fraction of strawberry yogurt cups (18/30) by 100%.

To find the percentage of yogurt cups that are not strawberry, you subtract the percentage of strawberry yogurt cups (60%) from 100%, which gives 40%.

Therefore, we can conclude we have correctly answered this question.

The complete question is the following:

Mrs. Dimitri purchased a case of yogurt cups. Out of every 10 yogurt cups, 6 are strawberry. There are 30 yogurt cups in the case. Choose True or False for each statement.

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Answer: The Tacos

Step-by-step explanation: This is because independent variable is the cause not the effect but i might be wrong because i don't have the answer options

The statement D. (x, y) ⇒ (x + 4, y – 2) shows the rule for the translation. Thus, option D is correct.

In geometry, a transformation is an operation that shifts, flips, or alters a shape to create a different one. A translation, which consistently and uniformly moves each point in a figure in one direction, is an example of a transformation. Translations are frequently referred to as slides.

For (x, y) => (x – 4, y – 2)  

x decreases by 4 and y decreases by 2 in other words, we may say that the graph shifts to the left by 4 units and downward by 2 units.

For (x, y) => (x – 4, y + 2)

x decreases by 4 and y increases by 2 in other words, we may say that the graph shifts to the left by 4 units and upward by 2 units.

For (x, y) => (x + 4, y + 2)

x increases by 4 and y increases by 2 in other words, we may say that the graph shifts to the right by 4 units and upward by 2 units.

For (x, y) => (x + 4, y – 2)

x increases by 4 and y decreases by 2 in other words, we may say that the graph shifts to the right by 4 units and downward by 2 units.

Thus, the correct option is option D.

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Answer:

the answer is D on edge

Step-by-step explanation:

picture below

An equation of a line parallel to that passes y = 3x + 1 through the point (-4, 1) is y = 3x + 13.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

Since the line is parallel to y = 3x + 1, the slope is equal to 3.

At data point (-4, 1) and a slope of 3, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 1 = 3(x - (-4))  

y - 1 = 3(x + 4)

y = 3x + 12 + 1

y = 3x + 13

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Answer: a) Line segment OA is a radius of the circle, and any other radius will have the same length as OA. So, line segments OB, OE, OF are also definitely equal in length to OA.

b) Triangle OAB is definitely isosceles because it has two equal sides, OA and OB, which are radii of the circle.

The vertex of the graph of this equation y = x² - x - 2 is equal to -2.25.

In this exercise, you are required to determine the vertex form of a quadratic function h(x) that is written in standard form. Mathematically, the vertex form of a quadratic equation is given by this formula:

y = a(x - h)² + k

Where:

For the given quadratic function y = x² - x - 2 in question 9, the axis of symmetry can be calculated as follows:

Axis of symmetry, Xmax = -b/2a

Axis of symmetry, Xmax = -(-1)/2(1)

Axis of symmetry, Xmax = 1/2

Axis of symmetry, Xmax = 1/2.

Next, we would determine the vertex when x = 1/2 as follows;

y = x² - x - 2

y = (1/2)² - 1/2 - 2

y = 1/4 - 1/2 - 2

y = 0.25 - 0.5 - 2

y = -2.25.

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Answer: a) The probability of matching the first five selected numbers to the later five numbers drawn is given by the formula:

P = (C(69,5) / C(69,5)) * (C(5,5) / C(26,1))

where C(n,r) is the number of combinations of n items taken r at a time. The first term in the product represents the number of ways to choose the first five numbers from the pool of 69, and the second term represents the probability of choosing the correct number from the second set of 26 numbers.

Simplifying the expression, we get:

P = (1) * (1/26) = 1/26

Therefore, the probability of matching the first five selected numbers to the later five numbers drawn is 1/26.

b) The probability of matching the sixth selected number to the later number drawn from the second set is given by:

P = C(1,1) / C(26,1) = 1/26

Therefore, the probability of matching the sixth selected number to the later number drawn is 1/26.

c) The probability of winning the jackpot is the product of the probabilities of matching the first five selected numbers to the later five numbers drawn and matching the sixth selected number to the later number drawn. Using the results from parts (a) and (b), we get:

P = (1/26) * (1/26) = 1/676

Therefore, the probability of winning the jackpot is 1 in 676.

Step-by-step explanation:

Answer:

[tex]\frac{9}{7}[/tex]

Step-by-step explanation:

[tex]\frac{6}{7}[/tex] ÷ [tex]\frac{2}{3}[/tex] = [tex]\frac{6}{7}[/tex] · [tex]\frac{3}{2}[/tex] = [tex]\frac{18}{14}[/tex] = [tex]\frac{9}{7}[/tex]

So, the answer is  [tex]\frac{9}{7}[/tex]

Answer: You can buy a maximum of 15 gallons of gas for $54.

To see why, you can divide the total amount of money you have by the cost per gallon:

$54 ÷ $3.60/gallon = 15 gallons

Step-by-step explanation:

The measure of BC is 110°

Semicircle is defined as a half circle formed by cutting the circle into two halves. It is formed when a line passes through the center and touches the two ends of the circle. This line is called the diameter of the circle.

line AB is the diameter of the circle and thus, AB gives a semi circle.

This means that, since angle Ina semi circle is 180°, the sum of AC and BC is 180°

therefore, 2x-30+x = 180°

3x = 180+30

3x = 210

divide both sides by 3

x = 210/3 = 70°

therefore BC = 2x-30

= 2×70-30

= 140-30

= 110

therefore the measure of BC is 110°

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Answer:

[tex] \frac{3}{5} = 0.6[/tex]

Step-by-step explanation:

By dividing the denominator and numerator by 4 you get:

[tex] \frac{12}{20 } = \frac{3}{5} [/tex]

Answer:

Step-by-step explanation:

The formula for the volume of a rectangular prism is V = lwh, where V is the volume, l is the length, w is the width, and h is the height.

We know that V = 5,890.625 cm^3, h = 14.5 cm, and l = 25 cm.

Substituting these values into the formula, we get:

5,890.625 = 25w(14.5)

5,890.625 = 362.5w

w = 16.25

Therefore, the width of the rectangular prism is 16.25 cm.

Answer:

It's A

Step-by-step explanation:
well all you have to do is divide like 5,890.625 dvided by 14.5 then divide again by 25 them bam you got 16.25

Answer:

It is not possible to tell from the graph whether k is greater than or less than zero. The graph only shows the shape of the function and the location of its intercepts and does not provide any information about the values of its parameters.

Answer:

Based on the graphed values of f(x) and independent variable x, the graph shows that C. k is greater than zero.

What is the value of k?

We have a c value of -1 which is where the y axis is intercepted.

We know that k isn't zero because the x and y values are not the same. We can also tell that k is not less than 0 because if it was, the f(x) for the smaller x value would lead to a larger f(x) value.

The only logical conclusion therefore, is that k is greater than zero.

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(a)   g(x) = 2f(x) + 3.

b.)

the inverse functions are:

f-1(x) = x/2

g¹(x) = (x - 3)/2

c.)

g¹(x) = 2*f-1(x+3).

d. ) we obtain the graph of g¹(x) = 2f-1(x+3). and  g(x) = 2f(x) + 3.

(b) The inverse function of f(x) = 2*x is f-1(x) = x/2.

We found the  inverse function of g(x) = 2f(x) + 3 = 2(x) + 3, by isolating f(x):

g(x) = 2*f(x) + 3

(g(x) - 3)/2 = f(x)

f-1(x) = (x - 3)/2

(d)   we work on this graph f-1(x) and g¹(x),  by graphing f(x) = 2x and y = x/2 on the same set of axes.

shift the graph of f(x) to the right by 3 units to obtain the graph of f-1(x+3), and stretch it vertically by a factor of 2 to obtain the graph of g¹(x) = 2f-1(x+3).

The resulting graph should have a horizontal asymptote at y = 3 and pass through the points (0, -1.5) and (6, 4.5).

To produce g(x) from f(x), we will use  a vertical stretch by a factor of 2, which results in 2f(x). Then, we shift the graph 3 units upward, which results in g(x) = 2f(x) + 3.

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Answer: 2/3q + 7/12

Step-by-step explanation: on khan academy

x=2,-13

Hope this helps

The statement "then x ∈ H implies x^n ∈ H for all n ≥ 0 and x^(-1) ∈ H" holds true in this case.

The term "abstract" refers to the fact that in abstract algebra, the focus is on studying algebraic structures in their own right, without necessarily referring to any specific set of numbers or operations. For example, instead of studying the arithmetic properties of the real numbers or the integers, abstract algebra studies the general properties of groups, which are sets equipped with a binary operation that satisfies certain axioms.

The statement "then x ∈ H implies[tex]x^n[/tex]∈ H for all n ≥ 0 and[tex]x^(-1)[/tex]∈ H" means that if x is an element of the set H, then x raised to any non-negative power (including 0) is also an element of H. Additionally, the inverse of x, denoted by [tex]x^(-1),[/tex] is also an element of H.

This statement is often used in the context of groups, where H is a subgroup of a larger group, and x is an element of that group. In this context, the statement means that if x is an element of H, then all powers of x (including the inverse) are also elements of H. This is because a subgroup is closed under the group operation, which in this case is multiplication (i.e., raising an element to a power).

For example, if H is the set of all positive integers, and x is the number 2, then[tex]x^2 = 4, x^3[/tex]= 8, and so on, are all elements of H. Additionally, the inverse of x, which is 1/x = 1/2, is also an element of H. Therefore, the statement "then x ∈ H implies[tex]x^n[/tex] ∈ H for all n ≥ 0 and[tex]x^(-1)[/tex]∈ H" holds true in this case.

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a) The sum of the measures of the interior angles of the regular nonagon is given by S₉ = 1260°

b) The measure of each interior angle of the regular nonagon is a = 140°

c) The measure of each exterior angle of the regular nonagon​ is b = 40°

Given data ,

Sum of the measures of the interior angles of a regular nonagon:

The Polygon Interior Angle Sum Theorem states that the sum of the measures of the interior angles of a polygon with n sides is given by the formula:

Sum of interior angles = (n - 2) * 180°

For a nonagon, which has 9 sides, we can plug in n = 9 into the formula:

Sum of interior angles = (9 - 2) * 180

Sum of interior angles = 7 * 180

Sum of interior angles = 1260°

So, the sum of the measures of the interior angles of a regular nonagon is 1260°

Measure of each interior angle of a regular nonagon:

Since a regular nonagon has 9 sides, we can divide the sum of the interior angles (which we calculated as 1260 degrees) by the number of sides to find the measure of each interior angle:

Measure of each interior angle = Sum of interior angles / Number of sides

Measure of each interior angle = 1260 / 9

Measure of each interior angle = 140°

So, the measure of each interior angle of a regular nonagon is 140°

Measure of each exterior angle of a regular nonagon:

The Polygon Exterior Angle Sum Theorem states that the sum of the measures of the exterior angles of a polygon with n sides is always 360°

Since a regular nonagon has 9 sides, we can divide 360 degrees by the number of sides to find the measure of each exterior angle:

Measure of each exterior angle = 360 / Number of sides

Measure of each exterior angle = 360 / 9

Measure of each exterior angle = 40°

Hence , the measure of each exterior angle of a regular nonagon is 40°

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