Algebra- 4

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factor:

(verb) to break up a polynomial into a product of polynomials that can be multiplied to get the original polynomial

root:

(noun) an x-value where the polynomial function intersects the x-axis; the x-values where f(x) = 0

zero of a function:

(noun) a value of the independent variable in a function that results in a value of zero for the dependent variable

zero product property :

(noun) the property stating that if a product is equal to zero, then at least one of the factors must be zero

Which statement describes the graph of f(x) = 4×2 + 20x + 25?

XX The graph intersects the x-axis at (2, 0) and (5, 0).

A function g(x) has x-intercepts at (1/2, 0) and (6, 0). Which could be g(x)?

XX g(x) = 2(x + 1)(x + 6)

The only solution of the equation x2 + bx + 16 = 0 is x = 4. What is the value of b?

b = -8

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

x = 4

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

(m + 5)(m – 9) = 0

Which equations are true for x = -2 and x = 2? Check all that apply.

a,d

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

x = -1 and x = ___

XX -3

Which function has zeros at x = 10 and x = 2?

f(x) = x2 – 12x + 20

Which equation is true for x = -6 and x = 2?

2×2 + 8x – 24 = 0

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

f(x) = x2 + 4x – 21

polynomial:

an expression involving a sum of powers in one or more variables multiplied by coefficients, where the powers must be whole numbers

viable :

capable of working successfully; practical; realistic; usable; possible

What are the solutions of the equation 4×2 + 3x = 24 – x?

-3 or 2

Two negative integers are 5 units apart on the number line, and their product is 126. What is the sum of the two integers?

-23

A rectangular piece of paper has a width that is 3 inches less than its length. It is cut in half along a diagonal to create two congruent right triangles with areas of 44 square inches. Which statements are true? Check all that apply.

The area of the rectangle is 88 square inches. The equation x2 – 3x – 88 = 0 can be used to solve for the length of the rectangle. The triangle has a base of 11 inches and a height of 8 inches.

Which is a solution to the equation?

(x −2)(x + 5) = 18

x = −7

Which is a solution to (x − 2)(x + 10) = 13?

x = 3

Which is a solution to (x – 3)(x + 9) = -27?

x = 0

What is the only solution of 2×2 + 8x = x2 – 16?

-4

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

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

13

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

-25

A rectangle has an area of 40 square units. The length is 6 units greater than the width.

10 by 4

Two negative integers are 8 units apart on the number line and have a product of 308.

Which equation could be used to determine x, the smaller negative integer?

x2 + 8x – 308 = 0

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

4 and -6

Two positive integers are 3 units apart on a number line. Their product is 108.

Which equation can be used to solve for m, the greater integer?

XX m(m + 3) = 108

approximate:

(verb) to come close to

isolate:

(verb) to separate from other substances; to place apart so as to be alone

What are the solution(s) to the quadratic equation 50 – x2 = 0?

x = ±5— 2 with a square root

What are the solutions of the quadratic equation (x + 3)2 = 49?

x = 4 and x = -10

Which are the roots of the quadratic function f(b) = b2 – 75? Check all that apply.

b = 5 sr 3 b = -5 sr 3

What are the solution(s) to the quadratic equation 9×2 = 4?

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

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

x = 4 and x = -4

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?

XX 8 -6 sr 2feet by 8 + 6 sr 2

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

adding 40 to both sides of the equation

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

x = 5 and x = -5

Which are the roots of the quadratic function f(q) = q2 – 125? Check all that apply.

XX q = 3 sr 5 q = -3 sr 5

Theo started to solve the quadratic equation (x + 2)2 – 9 = -5. He added 9 to both sides and the resulting equation was
(x + 2)2 = 4. Next, he took the square root of each side.

Which was the resulting equation of that step?

x + 2 = ±2

Which statement is true about the quadratic equation 8×2 − 5x + 3 = 0?

The leading coefficient is 8.

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

a

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

-7

What is the value of the discriminant for the quadratic equation 0 = 2×2 + x – 3?

25

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?

XX factoring out the constant

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

b

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

XX c

Determine the discriminant for the quadratic equation 0 = -2×2 + 3. Based on the discriminant value, how many real number solutions does the equation have?

Discriminant = b2 – 4ac

_______real number solutions

2

Which shows the correct substitution of the values a, b, and c from the equation 0 = 4×2 + 2x – 1 into the quadratic formula below?

a

Mattie uses the discriminant to determine the number of zeros the quadratic equation 0 = 3×2 – 7x + 4 has. Which best describes the discriminant and the number of zeros?

The equation has no zeros because the discriminant is not a perfect square.

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

1

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

8

What are the zeros of the function f(x) = x2 + 5x + 5 written in simplest radical form?

c

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

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

Rhett is solving the quadratic equation 0= x2 – 2x – 3 using the quadratic formula. Which shows the correct substitution of the values a, b, and c into the quadratic formula?

XX c

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

x = 5 squ 31

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

0

https://quizlet.com/83299681/solving-quadratic-equations-quadratic-formula-flash-cards/

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

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

What is the value of the greater number?

30

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

28

The product of two numbers is 450. The first number is half the second number. Which equation can be used to find x, the greater number?

1/2 x^2 = 450

he product of two positive numbers is 1,800. One number is twice the other number. The equation to determine the two numbers, with x representing the value of the lesser number, is x(2x) = 1,800.

What is the value of the lesser number?

30

A rocket is launched into the air. The projectile motion of the rocket can be modeled using h(t) = 112t -16t2, where t is the time since launch in seconds and h(t) is the height of the rocket at time t.

When will the rocket be 196 feet in the air?

The projectile motion of an object can be modeled using
s(t) = gt2 + v0t + s0, where g is the acceleration due to gravity, t is the time in seconds since launch, s(t) is the height after t seconds, v0 is the initial velocity, and s0 is the initial height. The acceleration due to gravity is -4.9 m/s2.

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?

XX s(t) = -4.9t2 + 39.2t + 39.2 s(t) = -4.9t2 + 39.2

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?

XX x(x + 9) = 713

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

A rectangle’s width is 2 meters shorter than its length, l. Its area is 168 square meters. Which equation can be used to find the length of the rectangle?

l(l – 2) = 168

Kyla makes a triangular school pennant. The area of the triangle is 180 square inches. The base of the pennant is z inches long. The height is 6 inches longer than twice the base length.

What is the height of the pennant? Recall the formula
A = 1/2bh.

30 inches

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?

XX x(x – 8) = 240 x2 – 8 = 240

The function g(x) is defined as g(x) = 6×2 + 23x – 4. When does g(x) = 0?

x = -4 or x = 1/6

A rectangle’s length is 5 inches more than twice its width. Its area is 50 square inches. Which equation can be used to find its width, w?

w(2w + 5) = 50

What are the solutions to m2 – 9 = 0?

m = -3 and m = 3

The first step in solving the quadratic equation -5×2 + 8 = 133 is to subtract__ from each side.

8

Mischa wrote the quadratic equation 0 = -x2 + 4x – 7 in standard form. What is the value of c in her equation?

-7

Between which two ordered pairs does the graph of f(x) = 1/2x^2 + x – 9 cross the negative x-axis?

(-6, 0) and (-5, 0)

What is the value of the discriminant, b2 − 4ac, for the quadratic equation 0 = −2×2 − 3x + 8, and what does it mean about the number of real solutions the equation has?

The discriminant is 73, so the equation has 2 real solutions.

Which is the graph of a quadratic equation that has a negative discriminant?

d

What is a solution to (x + 6)(x + 2) = 60?

x=4

What values of b satisfy 4(3b + 2)2 = 64?

b =1/2 and b = -2

Which equation has an a-value of -2, a b-value of 1, and a c-value of 3?

0 = -2×2 + x + 3

What is the value of the discriminant of the quadratic equation −2×2 = −8x + 8, and what does its value mean about the number of real number solutions the equation has?

The discriminant is equal to 0, which means the equation has one real number solution.

Which function has real zeros at x = −10 and x = −6?

f(x) = x2 + 16x + 60

Heather is writing a quadratic function that represents a parabola that touches but does not cross the x-axis at x = -6. Which function could Heather be writing?

f(x) = -x2 – 12x – 36

Liliana decides to crop a square photo 2 inches on each side to fit it into a frame. The area of the original photo was 121 square inches. In the equation (x + 2)2 = 121, x represents the side measure of the cropped photo.

What are the dimensions of the cropped photo?

9 inches by 9 inches

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