UPCAT Math 3 Algebra 1

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

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

B

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

b = -8

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

f(x) = x2 − 3x − 10

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

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

What are the solutions to x2 + 8x + 7 = 0?

x = -7 and x = -1

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

x2 - 4 = 0 4x2 = 16

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

f(x) = x2 - 12x + 20

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

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

4 and -6

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

-4

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

x = 3

Which is a solution to the equation?

(x −2)(x + 5) = 18

x = −7

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

x = 0

What are the solutions to the equation?

x2 + 6x = 40

x = −10 and x = 4

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

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

x = 4 and x = -4

What are the solutions to the quadratic equation 3(x − 4)2 = 75?

x = −1 and x = 9

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

x = 4 and x = -10

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(b) = b2 - 75? Check all that apply.

1and 2

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

1 and 2

What are the solution(s) of the quadratic equation 98 - x2 = 0?

C.

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

A.

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

144

Solve x2 + 14x = −24 by completing the square.

What is the solution set of the equation?

{−12, −2}

Solve x2 - 8x = 3 by completing the square. Which is the solution set of the equation?

A.

Which are the solutions of x2 = 19x + 1?

B.

What value of c makes x2 − 12x + c a perfect square trinomial?

36

Solve x2 = 12x - 15 by completing the square. Which is the solution set of the equation?

D.

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

9

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

D.

Atraeus is solving the quadratic equation by completing the square. What is the value of A.

7

What are the zeros of the quadratic function f(x)=2x2-10x-3

C

Sienne is solving the quadratic equation by completing the square. What is the value of A.

3

Patel is solving 8x2+16x+3=0 check all that apply

1,2,5

The first few steps in solving the quadratic equation 8x2+80x=-5 by completing the square are shown.

200

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 are the zeros of the quadratic function f(x) = 2x2 - 10x - 3?

c.

Isoke is solving the quadratic equation by completing the square.

10x2 + 40x - 13 = 0

10x2 + 40x = 13

A(x2 + 4x) = 13

What is the value of A?

10

Sunil is solving the quadratic equation by completing the square. What should sunil do next.

C.

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

Not b

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

Not d

What are the zeros of the quadratic function f(x)=16x2+32x-9

B.

Lie is solving a quadratic equation by completing the square. What should lie do next

B.

Which is a zero of the quadratic function f(x)=9x2-54x-19

Not b

In which step did yvonne make a mistake.

A.

In which step did tran first make an error.

Not d

What should gio do first.

A.

What should maya do first

C.

Which statement is true about the quadratic equation 8x2 − 5x + 3 = 0?

The leading coefficient is 8.

Determine the discriminant for the quadratic equation 0 = -2x2 + 3. Based on the discriminant value, how many real number solutions does the equation have?

Discriminant = b2 - 4ac

2

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

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

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

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

Discriminant = b2 - 4ac

not c

What are the values of a, b, and c in the quadratic equation 0 = 5x - 4x2 - 2?

a = -4, b = 5, c = -2

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

Quadratic formula: x =

not c or b

What is the value of the discriminant for the quadratic equation 0 = 2x2 + x - 3?

Discriminant = b2 - 4ac

not a

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

Quadratic formula:

not d

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 for the quadratic equation 0 = x + 2 + x2?

Discriminant = b2 - 4ac

-7

Which shows the correct substitution of the values a, b, and c from the equation -2 = -x + x2 - 4 into the quadratic formula?

Quadratic formula: x =

not c

A student is deriving the quadratic formula. Her first two steps are shown.

Step 1: -c = ax2 + bx

Step 2:

Which best explains or justifies Step 2?

factoring the binomial

Alessandro wrote the quadratic equation -6 = x2 + 4x - 1 in standard form. What is the value of c in his new equation?

not -1

Some of the steps in the derivation of the quadratic formula are shown.

Step 4:

Step 5:

Step 6:

Step 7:

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

not d

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

Quadratic formula:

a.

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

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

Some of the steps in the derivation of the quadratic formula are shown.

Step 3: -c + = a

Step 4a: -c + = a

Step 4b: + = a

Which best explains or justifies Step 4b?

converting to a common denominator

Anderson uses the discriminant to correctly find the number of real solutions of the quadratic equation x2 + 4x + 8 = 0. Which explanation could Anderson provide?

The equation has one real number solution because the discriminant is 0

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?

D.

Zacharias is using the quadratic formula to solve the equation 0 = -2x2 + 5x - 3. He begins by substituting as shown.

Quadratic formula: x = Substitution: x = What error did Zacharias make?

not a

Ramiya is using the quadratic formula to solve a quadratic equation. Her equation is x = after substituting the values of a, b, and c into the formula. Which is Ramiya's quadratic equation?

Quadratic formula: x =

0 = x2 + 3x + 2

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

not a

Quincy uses the quadratic formula to solve for the values of x in a quadratic equation. He finds the solution, in simplest radical form, to be x = .

Which best describes how many real number solutions the equation has?

not b or c

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

Quadratic formula: x =

not c

What is the discriminant of the quadratic equation 0 = 2x2 + 3x - 5?

not c

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

Quadratic formula: x =

c

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

A quadratic equation has a discriminant of 12. Which could be the equation?

0 = 2x2 + 6x + 3

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

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

not c or d

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

Quadratic formula: x =

not b

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

8

What is the discriminant of the quadratic equation 0 = 2x2 + 3x - 5?

not b or 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?

not 2 or 1

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

Quadratic formula: x =

not b

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?

not c

Mattie uses the discriminant to determine the number of zeros the quadratic equation 0 = 3x2 - 7x + 4 has. Which best describes the discriminant and the number of zeros?

The equation has two zeros because the discriminant is greater than 0.

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

f(x) = x2 + 16x + 60

What are the solutions to the quadratic equation 4(x + 2)2 = 36

x = −5 and x = 1

Solve x2 + 6x = 7 by completing the square. Which is the solution set of the equation?

{-7, 1}

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

x = 4

What are the zeros of the quadratic function f(x) = 2x2 + 8x - 3?

A.

Two numbers are 10 units away in different directions from their midpoint, m, on a number line. The product of the numbers is -99.

Which equation can be used to find m, the midpoint of the two numbers?

m2 - 100 = -99

Mariya is solving the quadratic equation by completing the square.

4x2 - 20x + 3 = 0

4x2 - 20x = -3

A(x2 - 5x) = -3

What is the value of A?

4

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

-7

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

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

D.

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

The first step in solving the quadratic equation -5x2 + 8 = 133 is to subtract from each side.

8