Quadratic Equations Practice Questions — SAT — Page 2

Question 21Question →

In the quadratic equation $x^2 - ax + b = 0$ , the roots $r$ and $s$ are positive integers. The quadratic equation $x^2 - (a+12)x + (b+12) = 0$ has roots $r^2$ and $s^2$ , where $a$ and $b$ are constants. What is the value of $a$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a

is 5.

Applying Vieta's formulas to both equations and using the algebraic identity

r^2 + s^2 = (r+s)^2 - 2rs

yields the system of equations

a^2 - 2b = a + 12

and

b^2 = b + 12

. Solving the second equation gives

b = 4

(discarding the negative root since

r, s > 0

). Substituting

b = 4

into the first equation yields

a^2 - a - 20 = 0

, which solves to

a = 5

(discarding

a = -4

since

r, s > 0

).

Step-by-Step Solution

1

Apply Vieta's formulas to the first quadratic equation.

r+s = a

and

rs = b

To express the sum and product of the roots of the first equation in terms of the coefficients

a

and

b

.

2

Apply Vieta's formulas to the second quadratic equation.

r^2+s^2 = a+12

and

r^2s^2 = b+12

To express the sum and product of the squared roots in terms of the coefficients of the second equation.

3

Express

r^2+s^2

in terms of

a

and

b

using the algebraic expansion of a binomial square.

a^2 - 2b = a+12

By rewriting

r^2+s^2

as

(r+s)^2 - 2rs

, we can substitute

a

and

b

to form a system of equations.

4

Solve for the constant

b

using the product relation.

b = 4

The relation

(rs)^2 = b^2 = b+12

simplifies to the quadratic equation

b^2 - b - 12 = 0

. Factoring yields

b = 4

or

b = -3

. Since

r

and

s

are positive integers, their product

b

must be positive, meaning

b = 4

.

5

Substitute

b = 4

into the sum relation to solve for

a

.

a = 5

Substituting

b = 4

yields

a^2 - 2(4) = a+12

, which simplifies to

a^2 - a - 20 = 0

. Factoring yields

a = 5

or

a = -4

. Since

r

and

s

are positive integers, their sum

a

must be positive, meaning

a = 5

.

Key Concept

Vieta's Formulas and Symmetric Polynomial Relationships

Estimated Time:3m 0s

Question 22Question →

In the quadratic equation $x^2 + bx + c = 0$ , $b$ and $c$ are integers such that $b + c = 10$ . If the equation has no real solutions, what is the maximum possible value of $c$ ?

18

19

14

6

Show answer & explanation

Answer: 18

Answer

The maximum possible value of

c

is 18.

For the quadratic equation to have no real solutions, the discriminant

b^2 - 4ac

must be negative. Given

a=1

, this simplifies to

b^2 - 4c < 0

. Using the constraint

b + c = 10

, we can express

c

as

10 - b

. Substituting this into the inequality gives

b^2 - 4(10 - b) < 0

, which simplifies to

b^2 + 4b - 40 < 0

. Solving the boundary equation

b^2 + 4b - 40 = 0

using the quadratic formula gives

b = -2 \pm \sqrt{44}

. Since

\sqrt{44} \approx 6.63

, the boundary values of

b

are approximately

-8.63

and

4.63

. Thus, the inequality holds for

-8.63 < b < 4.63

. To maximize

c = 10 - b

, we need the smallest integer value for

b

in this range, which is

b = -8

. Substituting

b = -8

gives the maximum value of

c = 10 - (-8) = 18

.

Step-by-Step Solution

1

State the condition for the quadratic equation to have no real solutions.

The discriminant must be negative:

b^2 - 4ac < 0

. Since

a = 1

, this simplifies to

b^2 - 4c < 0

.

A quadratic equation has no real solutions if and only if its discriminant is strictly less than zero.

2

Express

c

in terms of

b

using the given equation

b + c = 10

, and substitute it into the discriminant inequality.

Since

c = 10 - b

, substituting gives

b^2 - 4(10 - b) < 0

, which simplifies to

b^2 + 4b - 40 < 0

.

Substituting one variable in terms of the other allows us to solve a single-variable quadratic inequality.

3

Find the boundary values of

b

by solving the equation

b^2 + 4b - 40 = 0

using the quadratic formula.

b = \frac{-4 \pm \sqrt{4^2 - 4(1)(-40)}}{2} = \frac{-4 \pm \sqrt{176}}{2} = -2 \pm \sqrt{44}

. Since

\sqrt{44} \approx 6.63

, the boundaries are approximately

b \approx -8.63

and

b \approx 4.63

.

The roots of the quadratic equation define the boundaries of the interval that satisfies the inequality.

4

Determine the interval for

b

that satisfies the inequality and identify the value of

b

that maximizes

c

.

The inequality

b^2 + 4b - 40 < 0

is satisfied when

-8.63 < b < 4.63

. Since

c = 10 - b

, maximizing

c

requires selecting the smallest possible integer value of

b

in this range, which is

b = -8

.

Subtracting a smaller number from 10 results in a larger value for

c

.

5

Calculate the maximum value of

c

using the optimal value of

b

.

c = 10 - (-8) = 18

.

Substituting the minimum integer value of

b

gives the maximum integer value of

c

.

Key Concept

Analyzing the discriminant of a quadratic equation to determine the nature of its roots and using substitution to solve optimization problems under constraints.

Question 23Question →

If $z$ is a positive number such that $z^2 - 4z - 12 = 0$ , what is the value of $z$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

To find the positive value of

z

that satisfies the equation

z^2 - 4z - 12 = 0

, we factor the quadratic expression. Finding two numbers that multiply to

-12

and add to

-4

gives

-6

and

2

. Thus, the equation can be factored as

(z - 6)(z + 2) = 0

. Setting each factor to zero yields the solutions

z = 6

and

z = -2

. Since the problem states that

z

is a positive number, we discard the negative solution, leaving

6

as the final answer.

Step-by-Step Solution

1

Factor the quadratic equation

z^2 - 4z - 12 = 0

.

(z - 6)(z + 2) = 0

To solve the quadratic equation by finding two numbers that multiply to -12 and add to -4.

2

Solve for the roots of the equation.

z = 6

or

z = -2

Set each factor equal to zero and solve for

z

.

3

Apply the constraint that

z

must be positive.

z = 6

Discard the negative solution

z = -2

as the problem specifies

z

must be a positive number.

Key Concept

Solving quadratic equations by factoring

Question 24Question →

Which of the following is a solution to the equation $x^2 - 2x - 8 = 0$ ?

-4

2

4

-8

Show answer & explanation

Answer: 4

Answer

The correct solution is 4.

To find the solutions to the equation

x^2 - 2x - 8 = 0

, we factor the quadratic expression on the left side of the equation. We need two numbers that multiply to

-8

and add to

-2

. These numbers are

-4

and

2

. Thus, the equation can be rewritten in factored form as

(x - 4)(x + 2) = 0

. Setting each factor to zero yields the solutions

x = 4

and

x = -2

. Among the given options,

4

is the only correct solution.

Step-by-Step Solution

1

Factor the quadratic expression

x^2 - 2x - 8

.

The expression factors into

(x - 4)(x + 2)

.

We look for two numbers that multiply to the constant term

-8

and add to the linear coefficient

-2

. These numbers are

-4

and

2

.

2

Set each factor equal to zero to solve for

x

.

x - 4 = 0

or

x + 2 = 0

, which gives

x = 4

or

x = -2

.

By the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.

3

Identify the solution that appears in the options.

x = 4

is a solution.

Checking the given options,

4

is listed while

-2

is not.

Key Concept

Solving quadratic equations by factoring

Question 25Question →

If $x > 0$ and $x^2 - 3x - 18 = 0$ , what is the value of $x + 2$ ?

5

6

8

-1

Show answer & explanation

Answer: 8

Answer

8

Factoring the quadratic equation

x^2 - 3x - 18 = 0

yields

(x - 6)(x + 3) = 0

. This gives two solutions:

x = 6

and

x = -3

. Since the condition specifies

x > 0

, we choose

x = 6

. The question asks for the value of

x + 2

, which is

6 + 2 = 8

.

Step-by-Step Solution

1

Factor the quadratic equation

x^2 - 3x - 18 = 0

by finding two numbers that multiply to

-18

and add to

-3

.

(x - 6)(x + 3) = 0

Factoring allows us to find the individual roots of the quadratic equation.

2

Solve for the possible values of

x

by setting each factor equal to zero.

x = 6

or

x = -3

If the product of two factors is zero, then at least one of the factors must equal zero.

3

Apply the given constraint that

x > 0

.

x = 6

The problem specifies that

x

must be a positive number, so the negative root

x = -3

must be discarded.

4

Substitute the value of

x

into the expression

x + 2

.

8

The question asks for the value of

x + 2

, so we evaluate

6 + 2

.

Key Concept

Solving quadratic equations by factoring and applying constraints on roots

Estimated Time:1m 0s

Question 26Question →

If $(x - 4)^2 = 81$ and $x > 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 13

Answer

The value of

x

is 13.

Taking the square root of both sides of

(x - 4)^2 = 81

yields

x - 4 = 9

or

x - 4 = -9

. Solving these linear equations gives

x = 13

or

x = -5

. Since

x > 0

is specified, the correct value of

x

is 13.

Step-by-Step Solution

1

Take the square root of both sides of the equation.

x - 4 = 9

or

x - 4 = -9

Applying the square root property to solve the quadratic equation.

2

Solve each linear equation for

x

.

x = 13

or

x = -5

Adding 4 to both sides of each equation.

3

Apply the given constraint that

x > 0

.

x = 13

Since

-5

is not greater than 0, the only positive solution is 13.

Key Concept

Solving quadratic equations by taking square roots

Question 27Question →

In the quadratic equation $x^2 - 15x + k = 0$ , where $k$ is a constant, one of the real solutions is $4$ times the other solution. What is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

36

The correct answer is 36. By Vieta's formulas, the sum of the roots of the quadratic equation

x^2 - 15x + k = 0

is

15

, and the product of the roots is

k

. Let the roots be

r

and

4r

. Their sum is

r + 4r = 5r = 15

, which gives

r = 3

. The roots are

3

and

4(3) = 12

. The product of the roots is

3 \times 12 = 36

, which represents the constant term

k

.

Step-by-Step Solution

1

Express the relationship between the two roots.

Let the two roots be

r

and

4r

.

We are given that one of the real solutions is 4 times the other solution.

2

Use Vieta's formulas to set up an equation for the sum of the roots.

r + 4r = 15 \implies 5r = 15

For a quadratic equation in the form

x^2 - sx + p = 0

, the sum of the roots is equal to

s

.

3

Solve for the variable

r

and find both roots.

r = 3

, so the roots are

3

and

4(3) = 12

.

Dividing both sides of

5r = 15

by 5 gives

r = 3

. Substituting this back gives the two roots.

4

Calculate the product of the roots to find the value of

k

.

k = 3 \times 12 = 36

For a quadratic equation in the form

x^2 - sx + p = 0

, the product of the roots is equal to

p

, which in this case is

k

.

Key Concept

Using Vieta's formulas to find relationships between the coefficients and roots of a quadratic equation.

Question 28Question →

The quadratic equation $x^2 - (k + 3)x + (2k + 3) = 0$ has exactly one real solution, where $k$ is a constant. If $k > 1$ , what is the value of $k$ ?

-1

1

3

5

Show answer & explanation

Answer: 3

Answer

The value of

k

is

3

.

For the quadratic equation to have exactly one real solution, its discriminant must be equal to

0

. Setting the discriminant of

x^2 - (k + 3)x + (2k + 3) = 0

to

0

yields

[-(k + 3)]^2 - 4(1)(2k + 3) = 0

. Simplifying this yields

k^2 - 2k - 3 = 0

, which factors as

(k - 3)(k + 1) = 0

. This gives two potential values for

k

:

3

and

-1

. Because the problem specifies that

k > 1

, the value of

k

must be

3

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation and state the condition for having exactly one real solution.

The coefficients of the quadratic equation

x^2 - (k + 3)x + (2k + 3) = 0

are

a = 1

,

b = -(k + 3)

, and

c = 2k + 3

. For a quadratic equation to have exactly one real solution, its discriminant

b^2 - 4ac

must equal

0

.

A discriminant of zero indicates that the quadratic equation has two equal real roots, which is equivalent to exactly one distinct real solution.

2

Set up the discriminant equation in terms of

k

and simplify it.

[-(k + 3)]^2 - 4(1)(2k + 3) = 0

(k^2 + 6k + 9) - (8k + 12) = 0

k^2 - 2k - 3 = 0

This algebraic expansion and simplification transforms the discriminant condition into a solvable quadratic equation for

k

.

3

Solve the quadratic equation

k^2 - 2k - 3 = 0

by factoring.

(k - 3)(k + 1) = 0

, which gives

k = 3

or

k = -1

.

Factoring is the most direct method to find the roots of this quadratic equation.

4

Apply the constraint

k > 1

to select the correct value of

k

.

Since

k

must be greater than

1

, the value

k = -1

is discarded, leaving

k = 3

.

The solution must satisfy all given constraints in the problem statement.

Key Concept

Discriminant of a Quadratic Equation

Alternative Method

Instead of solving the quadratic equation algebraically, you can test the given choices for

k

. For the correct option of

3

, substituting

k = 3

into the original equation yields

x^2 - 6x + 9 = 0

. Factoring this expression gives

(x - 3)^2 = 0

, which clearly has exactly one real solution (

x = 3

). Testing the other options would not produce a perfect square trinomial.

Estimated Time:2m 0s

Question 29Question →

The profit $P(x)$ , in dollars, a company makes from selling $x$ units of a product is given by the function $P(x) = -3x^2 + bx - 1,500$ , where $b$ is a positive constant. If the maximum profit the company can make is $1,200$ dollars, what is the value of $b$ ?

Show answer & explanation

Answer: 180

Answer

180

The maximum profit of the quadratic profit function occurs at its vertex. The x-coordinate of the vertex of

P(x) = -3x^2 + bx - 1,500

is given by

x = -\frac{b}{2(-3)} = \frac{b}{6}

. Substituting this into the profit equation and setting it equal to

1,200

yields

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

. Simplifying this expression results in

\frac{b^2}{12} = 2,700

, which gives

b^2 = 32,400

. Since

b

is positive,

b = 180

.

Step-by-Step Solution

1

Identify the x-coordinate of the vertex of the quadratic function in terms of

b

.

The x-coordinate of the vertex is

x = \frac{b}{6}

.

A quadratic function in the form

f(x) = ax^2 + bx + c

has its maximum value at the vertex when the coefficient of the squared term,

a

, is negative. The x-coordinate of the vertex is given by

x = -\frac{b}{2a}

. For this function,

a = -3

, so the x-coordinate of the vertex is

x = -\frac{b}{2(-3)} = \frac{b}{6}

.

2

Substitute

x = \frac{b}{6}

back into the profit equation

P(x)

and set it equal to the maximum profit of

1,200

dollars.

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

The maximum profit of

1,200

dollars is the value of the function at the vertex.

3

Simplify the equation to solve for the positive constant

b

.

b = 180

Simplifying the term

-3\left(\frac{b}{6}\right)^2

gives

-3\left(\frac{b^2}{36}\right) = -\frac{b^2}{12}

. Simplifying the term

b\left(\frac{b}{6}\right)

gives

\frac{b^2}{6}

. Substituting these back in gives

-\frac{b^2}{12} + \frac{b^2}{6} - 1,500 = 1,200

. Combining the

b^2

terms gives

\frac{b^2}{12} - 1,500 = 1,200

. Adding

1,500

to both sides gives

\frac{b^2}{12} = 2,700

. Multiplying both sides by

12

yields

b^2 = 32,400

. Taking the square root of both sides gives

b = \pm 180

. Since

b

must be a positive constant,

b = 180

.

Key Concept

Finding the maximum value of a quadratic function by using the vertex formula

x = -\frac{b}{2a}

and evaluating the function at that point.

Question 30Question →

In the system of quadratic equations below, $c$ is an integer constant:

2x^2 - 11x + c = 0

x^2 - 15x + 4c = 0

If the first equation has two distinct real solutions and the second equation has no real solutions, what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

15

For the first equation

2x^2 - 11x + c = 0

to have two distinct real solutions, its discriminant must be positive:

\Delta_1 = (-11)^2 - 4(2)(c) = 121 - 8c > 0

, which gives

c < 15.125

. For the second equation

x^2 - 15x + 4c = 0

to have no real solutions, its discriminant must be negative:

\Delta_2 = (-15)^2 - 4(1)(4c) = 225 - 16c < 0

, which gives

c > 14.0625

. The only integer

c

that satisfies the inequality

14.0625 < c < 15.125

is

15

.

Step-by-Step Solution

1

Set up the inequality for the first equation's discriminant to have two distinct real solutions.

c < 15.125

A quadratic equation

ax^2 + bx + c = 0

has two distinct real solutions if its discriminant,

b^2 - 4ac

, is greater than zero. For

2x^2 - 11x + c = 0

, this means

(-11)^2 - 4(2)(c) > 0

, simplifying to

121 - 8c > 0

, or

c < 15.125

.

2

Set up the inequality for the second equation's discriminant to have no real solutions.

c > 14.0625

A quadratic equation has no real solutions if its discriminant is less than zero. For

x^2 - 15x + 4c = 0

, the discriminant is

(-15)^2 - 4(1)(4c) < 0

, simplifying to

225 - 16c < 0

, or

c > 14.0625

.

3

Find the integer value of

c

that satisfies both inequalities.

c = 15

Combining the results from the previous steps, we get

14.0625 < c < 15.125

. Since

c

is specified to be an integer, the only integer that lies in this open interval is

15

.

Key Concept

Using the discriminant to determine the number of real solutions of quadratic equations

Question 31Question →

In the quadratic equation $2x^2 - 11x + c = 0$ , $c$ is a constant. If the sum of the squares of the two real solutions to the equation is $\frac{85}{4}$ , what is the value of $c$ ?

9

18

-9

\frac{9}{2}

Show answer & explanation

Answer: 9

Answer

9

The sum of the solutions of the quadratic equation is

11/2

and the product of the solutions is

c/2

. The sum of the squares of the solutions is

(11/2)^2 - 2(c/2) = 121/4 - c

. Setting this equal to the given value of

85/4

and solving for

c

yields

c = 9

.

Step-by-Step Solution

1

Express the sum and product of the solutions using Vieta's formulas.

For the quadratic equation

2x^2 - 11x + c = 0

, the sum of the solutions is

x_1 + x_2 = -\frac{b}{a} = \frac{11}{2}

, and the product of the solutions is

x_1 x_2 = \frac{c}{a} = \frac{c}{2}

.

Vieta's formulas relate the coefficients of a polynomial to the sum and product of its roots.

2

Use the algebraic identity for the sum of squares of two variables.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

This identity allows us to express the sum of squares in terms of the sum and product of the roots.

3

Substitute the expressions from Step 1 into the identity and set it equal to the given sum of squares.

\left(\frac{11}{2}\right)^2 - 2\left(\frac{c}{2}\right) = \frac{85}{4} \Rightarrow \frac{121}{4} - c = \frac{85}{4}

To solve for

c

using the given value of the sum of squares, which is

\frac{85}{4}

.

4

Isolate the constant

c

.

c = \frac{121}{4} - \frac{85}{4} = \frac{36}{4} = 9

To find the numerical value of

c

.

Key Concept

Vieta's Formulas and Root Identities

Question 32Question →

In the quadratic equation $3x^2 + kx + 12 = 0$ , where $k$ is a constant, the difference between the two real solutions is $4$ . What is the value of $k^2$ ?

160

192

288

0

Show answer & explanation

Answer: 288

Answer

The correct value of

k^2

is 288.

The correct value is 288. The difference between the two solutions

r_1

and

r_2

of the quadratic equation

ax^2 + bx + c = 0

is given by

|r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting

a = 3

,

b = k

, and

c = 12

, we get

|r_1 - r_2| = \frac{\sqrt{k^2 - 4(3)(12)}}{3} = \frac{\sqrt{k^2 - 144}}{3}

. Setting this equal to the given difference of

4

yields

\frac{\sqrt{k^2 - 144}}{3} = 4

. Multiplying by 3 gives

\sqrt{k^2 - 144} = 12

, and squaring both sides gives

k^2 - 144 = 144

, which simplifies to

k^2 = 288

. Alternatively, using Viete's formulas,

r_1 + r_2 = -\frac{k}{3}

and

r_1 r_2 = 4

. Using the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we have

4^2 = (-\frac{k}{3})^2 - 4(4)

, which simplifies to

16 = \frac{k^2}{9} - 16

, leading to

\frac{k^2}{9} = 32

, or

k^2 = 288

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

3x^2 + kx + 12 = 0

.

The coefficients are

a = 3

,

b = k

, and

c = 12

.

These values are needed to apply the difference of roots formula or Viete's relations.

2

State the formula for the difference between the two solutions

r_1

and

r_2

of a quadratic equation.

|r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting the coefficients gives

|r_1 - r_2| = \frac{\sqrt{k^2 - 4(3)(12)}}{3} = \frac{\sqrt{k^2 - 144}}{3}

.

This formula relates the difference of the roots directly to the coefficients of the equation.

3

Set the expression for the difference of the roots equal to the given difference of

4

and solve for

k^2

.

\frac{\sqrt{k^2 - 144}}{3} = 4 \Rightarrow \sqrt{k^2 - 144} = 12 \Rightarrow k^2 - 144 = 144 \Rightarrow k^2 = 288

.

Squaring both sides and isolating

k^2

yields its value.

Key Concept

Difference of roots and discriminant of a quadratic equation

Alternative Method

Use Viete's relations: The sum of the roots is

r_1 + r_2 = -\frac{k}{3}

and the product of the roots is

r_1 r_2 = \frac{12}{3} = 4

. The difference between the roots is given as

|r_1 - r_2| = 4

. Square this relation to get

(r_1 - r_2)^2 = 16

. Expand and rewrite the identity as

(r_1 + r_2)^2 - 4r_1 r_2 = 16

. Substituting the sum and product, we get

(-\frac{k}{3})^2 - 4(4) = 16 \Rightarrow \frac{k^2}{9} - 16 = 16 \Rightarrow \frac{k^2}{9} = 32 \Rightarrow k^2 = 288

.

Estimated Time:2m 0s

Question 33Question →

A parabola in the $xy$ -plane has equation $y = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The parabola passes through the points $(2, 13)$ and $(8, 13)$ in the $xy$ -plane. If the minimum value of the quadratic function defined by this equation is $-5$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

The correct answer is 2. The axis of symmetry of the parabola is halfway between the x-coordinates of the two symmetric points

(2, 13)

and

(8, 13)

, which is

x = 5

. Since the minimum value of the function is

-5

, the vertex of the parabola is

(5, -5)

. Writing the equation in vertex form,

y = a(x - 5)^2 - 5

, and substituting the point

(2, 13)

yields

13 = a(2 - 5)^2 - 5

, which simplifies to

18 = 9a

, so

a = 2

.

Step-by-Step Solution

1

Find the axis of symmetry

x = h

using the symmetric points

(2, 13)

and

(8, 13)

.

h = 5

Since the points

(2, 13)

and

(8, 13)

have the same

y

-coordinate, the axis of symmetry of the parabola must lie exactly halfway between their

x

-coordinates:

h = \frac{2 + 8}{2} = 5

.

2

Determine the vertex

(h, k)

of the parabola.

(5, -5)

The minimum value of a quadratic function occurs at its vertex. The given minimum value is

-5

, which represents the

y

-coordinate (

k

) of the vertex. Combining this with

h = 5

, the vertex is

(5, -5)

.

3

Write the equation of the parabola in vertex form.

y = a(x - 5)^2 - 5

The vertex form of a quadratic equation is

y = a(x - h)^2 + k

. Substituting the values

h = 5

and

k = -5

yields this equation.

4

Substitute one of the given points into the equation to solve for

a

.

a = 2

Substituting

(2, 13)

into the equation gives

13 = a(2 - 5)^2 - 5 \Rightarrow 18 = 9a \Rightarrow a = 2

.

Key Concept

Finding the equation of a parabola using symmetry and vertex form

Estimated Time:2m 0s

Question 34Question →

For which of the following values of $k$ does the equation $\frac{x^2 - kx + 9}{x - 3} = 0$ have exactly one real solution?

6

3

-6

-3

Show answer & explanation

Answer: $-6$

Answer

The correct value is

-6

.

For the equation

\frac{x^2 - kx + 9}{x - 3} = 0

to have a solution, we must have

x \neq 3

. Multiplying both sides by the denominator yields the quadratic equation

x^2 - kx + 9 = 0

. For a quadratic equation to have exactly one real solution, its discriminant must equal zero:

(-k)^2 - 4(1)(9) = k^2 - 36 = 0

, which gives

k = 6

or

k = -6

. If

k = 6

, the equation is

(x-3)^2 = 0

, which has the single solution

x = 3

. Since

x = 3

is excluded by the domain restriction, the equation has zero real solutions. If

k = -6

, the equation is

(x+3)^2 = 0

, which has the single solution

x = -3

. Since

-3

is a valid value, this value of

k

yields exactly one real solution.

Step-by-Step Solution

1

Identify the domain restriction of the rational equation.

The denominator

x - 3

cannot be zero, which means

x \neq 3

.

Division by zero is undefined, so

x = 3

cannot be a solution to the equation.

2

Simplify the equation by multiplying both sides by the denominator.

x^2 - kx + 9 = 0

for

x \neq 3

.

A rational expression equals zero when its numerator is zero and its denominator is non-zero.

3

Set the discriminant of the quadratic equation to zero to find the values of

k

that yield exactly one real root.

\Delta = (-k)^2 - 4(1)(9) = k^2 - 36 = 0 \implies k = 6

or

k = -6

.

A quadratic equation has exactly one real solution (a double root) when its discriminant is zero.

4

Verify if the roots for each value of

k

are valid under the domain restriction

x \neq 3

.

If

k = 6

,

x^2 - 6x + 9 = (x - 3)^2 = 0 \implies x = 3

(invalid/extraneous). If

k = -6

,

x^2 + 6x + 9 = (x + 3)^2 = 0 \implies x = -3

(valid).

The value

k = 6

makes

x = 3

the only solution, which is excluded. Therefore, only

k = -6

yields exactly one valid real solution (

x = -3

).

Key Concept

Solving rational equations that simplify to quadratic equations, using the discriminant to find the number of solutions, and identifying extraneous solutions due to domain restrictions.

Estimated Time:2m 0s

Question 35Question →

If one of the solutions to the equation $2x^2 - 9x + c = 0$ , where $c$ is a constant, is $x = 2$ , what is the other solution to the equation?

-\frac{13}{2}

-\frac{5}{2}

\frac{5}{2}

7

Show answer & explanation

Answer: $\frac{5}{2}$

Answer

The other solution to the equation is

\frac{5}{2}

.

The correct answer is found by using the relationship between the coefficients of a quadratic equation and the sum of its roots. For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

. Substituting the values

a = 2

and

b = -9

, we find that the sum of the roots is

-\frac{-9}{2} = \frac{9}{2}

. Since one of the solutions is given as

2

, the other solution must be

\frac{9}{2} - 2 = \frac{5}{2}

. Alternatively, one can find the constant

c

by substituting

x = 2

into the equation to get

2(2)^2 - 9(2) + c = 0

, which yields

c = 10

. Solving the resulting equation

2x^2 - 9x + 10 = 0

by factoring gives

(2x - 5)(x - 2) = 0

, which confirms the other solution is

\frac{5}{2}

.

Step-by-Step Solution

1

Identify the relationship between the coefficients and the sum of the roots for a quadratic equation.

For any quadratic equation in the form

ax^2 + bx + c = 0

, the sum of its roots,

x_1

and

x_2

, is given by the formula

x_1 + x_2 = -\frac{b}{a}

.

This relationship allows us to find the second root directly without needing to calculate the value of the constant

c

first.

2

Substitute the given root and the coefficients of the quadratic equation into the sum of the roots formula.

Here,

a = 2

and

b = -9

. The given root is

x_1 = 2

. Substituting these values gives

2 + x_2 = -\frac{-9}{2}

, which simplifies to

2 + x_2 = \frac{9}{2}

.

This sets up a linear equation to solve for the unknown root,

x_2

.

3

Solve for the second root

x_2

.

x_2 = \frac{9}{2} - 2 = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}

.

This isolates the unknown variable to find the final value.

Key Concept

Using the sum of roots formula (

x_1 + x_2 = -\frac{b}{a}

) to solve for an unknown solution of a quadratic equation.

Question 36Question →

In the quadratic equation $x^2 + kx + (k + 3) = 0$ , $k$ is a constant. If the equation has exactly one real solution, which of the following is a possible value of $k$ ?

-6

3

6

2

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. For the quadratic equation

x^2 + kx + (k + 3) = 0

to have exactly one real solution, its discriminant must be equal to zero. Setting the discriminant

b^2 - 4ac

to zero gives

k^2 - 4(1)(k + 3) = 0

, which simplifies to

k^2 - 4k - 12 = 0

. Factoring this equation yields

(k - 6)(k + 2) = 0

, meaning the possible values of

k

are

6

and

-2

. Among the choices, 6 is the only possible value listed.

Step-by-Step Solution

1

Identify the coefficients of the given quadratic equation

x^2 + kx + (k + 3) = 0

.

The coefficients are

a = 1

,

b = k

, and

c = k + 3

.

To apply the discriminant formula, we need to identify the standard form coefficients

a

,

b

, and

c

.

2

Set the discriminant of the quadratic equation to zero.

The discriminant is

D = b^2 - 4ac

. Setting

D = 0

gives

k^2 - 4(1)(k + 3) = 0

, which simplifies to

k^2 - 4k - 12 = 0

.

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Solve the quadratic equation

k^2 - 4k - 12 = 0

for

k

.

Factoring the quadratic yields

(k - 6)(k + 2) = 0

. Therefore, the possible values of

k

are

6

and

-2

.

Solving the equation reveals the values of the constant

k

that satisfy the condition.

Key Concept

Using the discriminant of a quadratic equation to determine the number of real solutions.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

k^2 - 4k - 12 = 0

, where

k = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)} = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm 8}{2}

, giving

k = 6

or

k = -2

.

Estimated Time:1m 30s

Question 37Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are constants, has its vertex at $(4, 9)$ . What is the positive difference between the two $x$ -intercepts of the graph?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

The quadratic function can be represented in vertex form as

f(x) = -(x - 4)^2 + 9

since the leading coefficient is

-1

and the vertex is at

(4, 9)

. Setting the function equal to zero to find the

x

-intercepts gives

(x - 4)^2 = 9

, which yields

x = 7

and

x = 1

. The positive difference between these intercepts is

7 - 1 = 6

.

Step-by-Step Solution

1

Write the quadratic function in vertex form using the given vertex

(4, 9)

and the leading coefficient.

f(x) = -(x - 4)^2 + 9

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex. Since the coefficient of

x^2

is

-1

, we have

a = -1

,

h = 4

, and

k = 9

.

2

Set

f(x) = 0

to find the

x

-intercepts.

(x - 4)^2 = 9

The

x

-intercepts of a graph are the points where

f(x) = 0

.

3

Solve for

x

.

x = 7

and

x = 1

Taking the square root of both sides gives

x - 4 = \pm 3

, which results in

x = 7

and

x = 1

.

4

Find the positive difference between the two

x

-intercepts.

6

Subtract the smaller

x

-intercept from the larger

x

-intercept:

7 - 1 = 6

.

Key Concept

Finding the intercepts of a quadratic function using its vertex form

Alternative Method

Alternatively, expand the vertex form

f(x) = -(x - 4)^2 + 9

to get

f(x) = -(x^2 - 8x + 16) + 9 = -x^2 + 8x - 7

. Factoring this expression gives

f(x) = -(x - 7)(x - 1)

. The roots are

x = 7

and

x = 1

, and their difference is

7 - 1 = 6

.

Estimated Time:1m 30s

Question 38Question →

In the quadratic equation $x^2 - 8x + k = 0$ , $k$ is a constant. If the difference between the two real solutions to the equation is $2$ , what is the value of $k$ ?

15

-15

12

8

Show answer & explanation

Answer: 15

Answer

15

The correct answer is 15. The sum of the solutions to the quadratic equation

x^2 - 8x + k = 0

is given by

-\frac{-8}{1} = 8

. We are given that the difference between the two solutions is

2

. Let the two solutions be

x_1

and

x_2

. We can set up the system of equations:

x_1 + x_2 = 8

and

x_1 - x_2 = 2

. Adding the equations gives

2x_1 = 10

, so

x_1 = 5

. Thus,

x_2 = 3

. The product of the solutions is equal to the constant term

k

. Therefore,

k = x_1 \times x_2 = 5 \times 3 = 15

. Alternatively, using the difference of roots formula,

\frac{\sqrt{b^2 - 4ac}}{a} = 2 \Rightarrow \frac{\sqrt{(-8)^2 - 4(1)(k)}}{1} = 2 \Rightarrow \sqrt{64 - 4k} = 2

. Squaring both sides gives

64 - 4k = 4 \Rightarrow 4k = 60 \Rightarrow k = 15

.

Step-by-Step Solution

1

Find the sum of the two solutions of the quadratic equation.

The sum of the solutions is

8

.

For a quadratic equation in the form

x^2 + bx + c = 0

, the sum of the solutions is equal to

-b

. Here,

-(-8) = 8

.

2

Set up a system of linear equations using the sum and the given difference of the solutions to find the individual solutions.

The solutions are

5

and

3

.

Let the solutions be

x_1

and

x_2

. We have the system:

x_1 + x_2 = 8

and

x_1 - x_2 = 2

. Adding these two equations gives

2x_1 = 10

, so

x_1 = 5

. Substituting

5

back into the first equation gives

5 + x_2 = 8

, so

x_2 = 3

.

3

Use the product of the solutions to determine the value of

k

.

The value of

k

is

15

.

For a quadratic equation in the form

x^2 + bx + c = 0

, the product of the solutions is equal to

c

, which is

k

. Therefore,

k = x_1 \times x_2 = 5 \times 3 = 15

.

Key Concept

Relationship between the roots and coefficients of a quadratic equation (Vieta's Formulas).

Question 39Question →

A projectile is launched from the ground. Its height, in feet, $t$ seconds after launch is modeled by the function $h(t) = -16t^2 + v_0 t$ , where $v_0$ is the initial upward velocity in feet per second. If the projectile reaches its maximum height of $144$ feet, what is the value of $v_0$ ?

Show answer & explanation

Answer: 96

Answer

The correct answer is 96. The initial velocity of the projectile must be 96 feet per second to reach a maximum height of 144 feet.

The maximum height of a projectile modeled by a quadratic function is the

y

-value of its vertex. For

h(t) = -16t^2 + v_0 t

, the time

t

at the vertex is given by

t = -\frac{b}{2a} = -\frac{v_0}{2(-16)} = \frac{v_0}{32}

. Substituting this value of

t

back into the height equation yields the maximum height:

h\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0^2}{1024}\right) + \frac{v_0^2}{32} = -\frac{v_0^2}{64} + \frac{v_0^2}{32} = \frac{v_0^2}{64}

. Given that the maximum height is

144

feet, we set

\frac{v_0^2}{64} = 144

, which simplifies to

v_0^2 = 144 \times 64

. Taking the square root of both sides gives

v_0 = 12 \times 8 = 96

.

Step-by-Step Solution

1

Find the time

t

at which the maximum height occurs using the vertex formula.

t = \frac{v_0}{32}

For a quadratic function of the form

f(x) = ax^2 + bx + c

, the vertex occurs at

x = -\frac{b}{2a}

. Here,

a = -16

and

b = v_0

, so the vertex is at

t = -\frac{v_0}{2(-16)} = \frac{v_0}{32}

.

2

Substitute the time

t = \frac{v_0}{32}

into the height function to express the maximum height in terms of

v_0

.

h\left(\frac{v_0}{32}\right) = \frac{v_0^2}{64}

Evaluating

h\left(\frac{v_0}{32}\right)

gives

-16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0^2}{1024}\right) + \frac{v_0^2}{32} = -\frac{v_0^2}{64} + \frac{v_0^2}{32} = \frac{v_0^2}{64}

.

3

Set the maximum height expression equal to the given maximum height of

144

feet and solve for

v_0

.

v_0 = 96

Setting

\frac{v_0^2}{64} = 144

yields

v_0^2 = 144 \times 64 = 9216

. Taking the positive square root (since

v_0

represents a positive initial upward velocity) gives

v_0 = \sqrt{144} \times \sqrt{64} = 12 \times 8 = 96

.

Key Concept

Determining the vertex coordinates of a quadratic function to find maximum value in context.

Question 40Question →

The daily profit, in dollars, of a small company is modeled by the quadratic function $P(x) = -2x^2 + 120x - 1000$ , where $x$ represents the number of items the company produces and sells each day. For what number of items produced and sold, greater than $20$ , will the company break even (meaning its daily profit is $\$0$ )?

10

30

40

50

Show answer & explanation

Answer: 50

Answer

The company breaks even when it produces and sells 50 items.

To find the break-even points, set the profit function to zero:

-2x^2 + 120x - 1000 = 0

. Dividing both sides by

-2

yields

x^2 - 60x + 500 = 0

. Factoring this equation gives

(x - 10)(x - 50) = 0

, which results in the solutions

x = 10

and

x = 50

. Since the question specifies that the number of items must be greater than

20

, the correct answer is 50.

Step-by-Step Solution

1

Set the profit function equal to zero to find the break-even points.

-2x^2 + 120x - 1000 = 0

Breaking even means the profit,

P(x)

, is equal to 0.

2

Divide the entire equation by the common factor of

-2

to simplify the quadratic expression.

x^2 - 60x + 500 = 0

Simplifying the quadratic expression makes it easier to factor.

3

Factor the quadratic equation by finding two numbers that multiply to

500

and add to

-60

.

(x - 10)(x - 50) = 0

Factoring allows us to find the individual roots of the equation.

4

Solve for

x

by setting each factor equal to zero.

x = 10

or

x = 50

The zero product property states that if a product of factors is zero, at least one factor must be zero.

5

Apply the constraint that the number of items must be greater than

20

.

x = 50

Out of the two roots, only 50 is greater than 20.

Key Concept

Solving quadratic equations in real-world contexts by finding roots and applying constraints.

Quadratic Equations

Answer

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Key Concept

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Alternative Method

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