Algebra I: Quadratic Equations and Absolute Value

What is the vertex of the absolute value function below?

(-3, 2)

What is the vertex of f(x) = |x + 8| - 3?

(-8,-3)

What is the domain of the absolute value function below?

(-infinity, infinity)

The function g(x) = x2 - 10x + 24 is graphed on a coordinate plane. Where will the function cross the x-axis?

(4, 0) and (6, 0)

An equation has solutions of m = -5 and m = 9. Which could be the equation?

(m+5)(m-9)=0

Which are the solutions of x2 = -11x + 4?

-11 - square root 137/2, -11 + square root 137/2

Which are the solutions of x2 = -13x - 4?

-13 - square root 153/2, -13+ square root 153/2

The product of two consecutive negative integers is 600. What is the value of the lesser integer?

-25

Which are the solutions of x2 = -5x + 8?

-5- square root 57/2, -5+ square root 57/2

What is the discriminant of the quadratic equation 3 - 4x = -6x2?

-56

What is the value of the discriminant for the quadratic equation 0 = x + 2 + x2?

-7

A quadratic equation has exactly one real number solution. Which is the value of its discriminant?

0

A quadratic equation of the form 0=ax2+bx+c has a discriminant value of -16. How many real number solutions does the equation have?

0

Determine the discriminant for the quadratic equation -3=x2+4x+1. Based on the discriminant value, how many real number solutions does the equation have?

1

What is the positive solution to the equation 0 = -x2 + 2x + 1?

1+ the square root of 2

Two factors of -48 have a difference of 19. The factor with a greater absolute value is positive. What is the sum of the factors?

13

Sergei buys a rectangular rug for his living room. He measures the diagonal of the rug to be 18 feet. The length of the rug is 3 feet longer than the width. What are the approximate dimensions of the rug? Round each dimension to the nearest tenth of a foot.

14.1 feet by 11.1 feet

What value of c makes x2 − 24x + c a perfect square trinomial?

144

What is the value of the discriminant for the quadratic equation -3 = -x2 + 2x?

16

What number should be added to both sides of the equation to complete the square? x2 + 8x = 4

16

Jillian is trying to save water, so she reduces the size of her square grass lawn by 8 feet on each side. The area of the smaller lawn is 144 square feet. In the equation(x - 8)2 = 144, x represents the side measure of the original lawn. What were the dimensions of the original lawn?

20 feet by 20 feet

The first few steps in solving the quadratic equation 8x2 + 80x = −5 by completing the square are shown. 8x2 + 80x = −5 8(x2 + 10x) = −5 8(x2 + 10x + 25) = −5 + ______ Which number is missing in the last step?

200

The product of two consecutive positive integers is 812. What is the value of the lesser integer?

28

Sienna is solving the quadratic equation by completing the square. 3x2 + 9x - 4 = 0 3x2 + 9x = 4 A(x2 + 3x) = 4 What is the value of A?

3

Which of the following is the value of a when the function f(x)=3|x| is written in the standard form of an absolute value function?

3

An object is dropped from a bridge and allowed to freefall to the ground. The height of the object over time can be modeled using h(t)=144-16t2. How many seconds will it take the object to reach the ground?

3 seconds

The product of two positive numbers is 750. The first number is 5 less than the second number. The equationx(x - 5) = 750 can be used to find x, the value of the greater number. What is the value of the greater number?

30

For what values of x is x2 + 2x = 24 true?

4 and -6

A rectangle's length is 2 units more than twice its width. Its area is 40 square units. The equation w(2w + 2) = 40 can be used to find w, the width of the rectangle. What is the width of the rectangle?

4 units

The standard diameter of a golf ball is 42.67 mm. A golf ball factory does quality control on the balls it manufactures. Golf balls are randomly measured to ensure the correct size. One day, an inspector decides to stop production if the discrepancy in diameter is more than 0.002 mm. What is the range of acceptable values?

42.670 mm to 42.672 mm

Atraeus is solving the quadratic equation by completing the square. 7x2 - 14x + 6 = 0 7x2 - 14x = -6 A(x2 - 2x) = -6 What is the value of A?

7

What is the discriminant of the quadratic equation 0 = -x2 + 4x - 2?

8

What is the positive solution of x2 - 36 = 5x?

9

What number should be added to both sides of the equation to complete the square? x2 - 6x = 5

9

What value of c makes x2 + 6x + c a perfect square trinomial?

9

The first few steps in solving the quadratic equation 9x2 + 49x = 22 − 5x by completing the square are shown. 9x2 + 49x = 22 − 5x 9x2 + 54x = 22 9(x2 + 6x) = 22 Which is the best step to do next to solve the equation by completing the square?

9(x2+6x+9)=103

What are the zeros of the quadratic function f(x) = 6x2 + 12x - 7?

A

Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options.

A B E

Which are the solutions of x2 = -7x - 8?

B

Which shows the correct substitution of the values a, b, and c from the equation 1 = -2x + 3x2 + 1 into the quadratic formula?

B

What are the zeros of the quadratic function f(x) = 8x2 - 16x - 15?

C

Which shows the correct substitution of the values a, b, and c from the equation 0 = - 3x2 - 2x + 6 into the quadratic formula?

C

What are the zeros of the quadratic function f(x) = 2x2 + 16x - 9?

D

Sunil is solving the quadratic equation by completing the square. 6x2 + 24x + 5 = 0 6x2 + 24x = -5 What should Sunil do next?

Factor 6 out of the variable terms

Which of the following is the graph of f(x)=|x| translated 2 units right, 2 units up, and dilated by a factor of 1/3?

Graph C

Gio is solving the quadratic equation by completing the square. 5x2 + 15x - 4 = 0 What should Gio do first?

Isolate the constant first

What is the first step in solving the quadratic equation x2 = 9/16?

Take the square root of both sides of the equation

What is the value of the discriminant, b2 − 4ac, for the quadratic equation 0 = x2 − 4x + 5, and what does it mean about the number of real solutions the equation has?

The discriminant is -4, so the equation has no real solutions

What is the value of the discriminant of the quadratic equation −1 = 5x2 −2x, and what does its value mean about the number of real number solutions the equation has?

The discriminant is equal to −16, which means the equation has no real number solutions

Which statement is true about f(x) = -6|x + 5| - 2?

The graph of f(x) is a horizontal compression of the graph of the parent function.

Which statement describes the graph of f(x) = 4x2 + 20x + 25?

The graph touches the x-axis at (-2.5, 0)

Which best explains why the expression cannot be rewritten as during the next step?

The square root of terms separated by addition and subtraction cannot be calculated individually

Quincy uses the quadratic formula to solve for the values of x in a quadratic equation. He finds the solution, in simplest radical form. Which best describes how many real number solutions the equation has?

Zero, because the discriminant is negative

What are the values of a, b, and c in the quadratic equation -2x2 + 4x - 3 = 0?

a=-2, b=4, c=-3

What are the values of a, b, and c in the quadratic equation 0 = 1/2x2 - 3x - 2?

a=1/2, b=-3, c=-2

What is the first step in solving the quadratic equation x2 - 40 = 0?

adding 40 to both sides of the equation

Which are the roots of the quadratic function f(b) = b2 - 75? Select two options.

b=5 square root 3 b=-5 square root 3

What are the domain and range of f(x) = 2|x - 4|?

domain:(-infinity, infinity) ; range: f(x)>0

Which function has a vertex at (2, 6)?

f(x) = 2|x - 2| + 6

Which function has real zeros at x = 3 and x = 7?

f(x) = x^2 - 10x + 21

Which function has zeros at x = −2 and x = 5?

f(x) = x^2 - 3x - 10

Which of the following is the parent function of all absolute value functions?

f(x)=|x|

Bao Yu is completing the square to solve the polynomial equation. x2 - x - 3 = 0 What should be her first step?

isolating the constant, -3

A rocket is launched from the ground at an initial velocity of 39.2 meters per second. Which equation can be used to model the height of the rocket after t seconds?

s(t) = -4.9t2 + 39.2t

Aisha wants to make two quilts, each with the same area. The first quilt will be square with sides s feet long. The second quilt will be a rectangle with a width that is half the length of a side of the square quilt and a length that is 6 feet longer than a side length of the square quilt. Which quadratic equation can be used to find s, the side length of the square quilt?

s2=(1/2s)(s+6)

The first step for deriving the quadratic formula from the quadratic equation, 0 = ax2 + bx + c, is shown. Step 1: -c = ax2 + bx Which best explains or justifies Step 1?

subtraction property of equality

John has 48 square centimeter tiles he wants to use to create a mosaic. He wants the mosaic to be rectangular with a length that is 2 centimeters longer than the width. Which equation could John solve to find w, the greatest width in centimeters he can use for the mosaic?

w(w+2) = 48

What are the solution(s) to the quadratic equation 40 − x2 = 0?

x = ±2 square root 10

The product of two numbers is 240. The first number is 8 less than the second number. Which equation can be used to find x, the lesser number?

x(x+8)=240

Two positive integers have a product of 176. One integer is 5 less than the other integer. Which equation can be used to find the value of x, the greater integer?

x(x-5) = 176

The product of two numbers is 713. The first number is 9 more than the second number. Which equation can be used to find x, the greater number?

x(x-9)=713

The roots of the function f(x) = x2 - 2x - 3 are shown. What is the missing number?

x=-1 and x=3

What are the solutions to the equation? x2 + 6x = 40

x=-10 and x=4

What are the solutions to the equation 0 = x2 - x -6? Select two options

x=-2 x=3

In simplest radical form, what are the solutions to the quadratic equation 0 = -3x2 - 4x + 5?

x=-2+ the square root of 19 divided by 3

What are the solutions to the quadratic equation 4x2 = 64?

x=-4 and x=4

Nathaniel is using the quadratic formula to solve 0 = x2 + 5x - 6. His steps are shown below. What are the solutions to the equation?

x=-6, 1

Which is a solution to the equation? (x −2)(x + 5) = 18

x=-7

What are the solutions of 3x2 + 14x + 16 = 0?

x=-8/3, -2

A computer company makes a rectangular screen with a diagonal of 20 inches. The width of the screen is 4 inches less than its length. The dimensions of the computer screen are modeled by the equation x2 + (x - 4)2 = 202. What is the value of x, the length of the screen?

x=16

What are the solution(s) to the quadratic equation 9x2 = 4?

x=2/3 and x=-2/3

Blake knows that one of the solutions to x2 - 6x + 8 = 0 is x = 2. What is the other solution?

x=4

What are the solutions to the quadratic equation x2 - 16 = 0?

x=4 and x=-4

One of the solutions to x2 − 2x − 15 = 0 is x = −3. What is the other solution?

x=5

What are the solution(s) to the quadratic equation x2 - 25 = 0?

x=5 and x=-5

In simplest radical form, what are the solutions to the quadratic equation 6 = x2 - 10x?

x=5+ the square root of 31

A square picture with a side length of 4 inches needs to be enlarged. The final area needs to be 81 square inches. Which equation can be used to solve for x, the increase in side length of the square in inches?

x^2 + 8x - 65 = 0

The product of two consecutive integers is 420. An equation is written in standard form to solve for the smaller integer by factoring. What is the constant of the quadratic expression in this equation?

x^2 + x + -420 = 0

What are the solutions to the quadratic equation (5y + 6)2 = 24?

y=-6+2 square root 6/5 and y=-6-2 square root 6/5

Solve x2 + 14x = −24 by completing the square. What is the solution set of the equation?

{-12, -2}