Quadratic Equations Practice Questions — SAT — Page 3

Question 41Question →

In the quadratic equation $2x^2 - 12x + k = 0$ , $k$ is a constant. If the sum of the squares of the solutions to the equation is $26$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The correct answer is

10

. By expressing the sum of the squares of the solutions as

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

, we can substitute the sum of the solutions (

-\frac{-12}{2} = 6

) and the product of the solutions (

\frac{k}{2}

) directly into the expression. This gives

26 = 36 - k

. Solving for

k

yields

k = 10

. Alternatively, solving the quadratic equation using the quadratic formula yields solutions

3 + \frac{\sqrt{144-8k}}{4}

and

3 - \frac{\sqrt{144-8k}}{4}

. Squaring these solutions and setting their sum equal to

26

simplifies to

18 + 2\left(\frac{144-8k}{16}\right) = 26

, which also solves to

k = 10

.

Step-by-Step Solution

1

Find the sum and product of the solutions using the coefficients of the quadratic equation.

The sum of the solutions is

6

and the product of the solutions is

\frac{k}{2}

.

By Vieta's formulas, for any quadratic equation

ax^2 + bx + c = 0

with solutions

x_1

and

x_2

, the sum of the solutions is

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

and the product of the solutions is

x_1 x_2 = \frac{c}{a}

.

2

Apply the algebraic identity to express the sum of the squares of the solutions in terms of their sum and product.

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

This identity allows us to substitute the sum and product of the solutions directly without solving for the individual solutions first.

3

Substitute the values and solve for

k

.

26 = 6^2 - 2\left(\frac{k}{2}\right) \implies 26 = 36 - k \implies k = 10

Substituting the given sum of squares (

26

), the sum of solutions (

6

), and the product of solutions (

\frac{k}{2}

) allows us to solve for the unknown constant

k

directly.

Key Concept

Using the relationship between the roots and coefficients of a quadratic equation (Vieta's formulas) in combination with algebraic identities to solve for unknown constants.

Question 42Question →

In the quadratic equation $2x^2 - bx + 18 = 0$ , $b$ is a positive constant. If one of the solutions to the equation is $4$ times the other solution, what is the value of $b$ ?

Show answer & explanation

Answer: 15

Answer

15

By Vieta's formulas, the product of the roots of the quadratic equation

2x^2 - bx + 18 = 0

is

\frac{18}{2} = 9

. Letting the roots be

r_1

and

r_2

with

r_1 = 4r_2

, we have

4r_2^2 = 9

, which yields

r_2 = \frac{3}{2}

(since

b > 0

implies the roots must be positive). Thus,

r_1 = 6

. The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and by Vieta's formulas, this sum equals

\frac{b}{2}

. Solving for

b

gives

15

.

Step-by-Step Solution

1

Set up the relationships for the product and sum of the roots using Vieta's formulas.

r_1 \cdot r_2 = 9

and

r_1 + r_2 = \frac{b}{2}

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

2

Substitute the given condition that one root is

4

times the other (

r_1 = 4r_2

) into the product equation.

4r_2^2 = 9

This reduces the product equation to a single variable equation in terms of

r_2

.

3

Solve for

r_2

and determine its sign based on the constraint that

b

is positive.

r_2 = \frac{3}{2}

Since

b > 0

, the sum of the roots

5r_2 = \frac{b}{2}

must be positive, which requires

r_2 > 0

.

4

Calculate the second root

r_1

and then use the sum of the roots to find

b

.

b = 15

The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and since

r_1 + r_2 = \frac{b}{2}

, we have

\frac{b}{2} = \frac{15}{2}

.

Key Concept

Relationship between roots and coefficients of a quadratic equation

Question 43Question →

In the quadratic equation $x^2 - px + q = 0$ , $p$ and $q$ are positive constants. If the equation has exactly one real solution, what is the value of $\frac{p^2}{q}$ ?

-4

4

2

16

Show answer & explanation

Answer: $4$

Answer

The correct answer is

4

.

The correct answer is

4

. For any quadratic equation of the form

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant

\Delta = b^2 - 4ac

must equal

0

. In the equation

x^2 - px + q = 0

, we identify

a = 1

,

b = -p

, and

c = q

. Substituting these values into the discriminant formula gives

(-p)^2 - 4(1)(q) = 0

, which simplifies to

p^2 - 4q = 0

. Rearranging this equation gives

p^2 = 4q

. Since

q

is a positive constant, we can divide both sides of the equation by

q

to isolate the ratio, yielding

\frac{p^2}{q} = 4

.

Step-by-Step Solution

1

Recall the condition for a quadratic equation of the form

ax^2 + bx + c = 0

to have exactly one real solution.

The discriminant,

\Delta = b^2 - 4ac

, must be equal to

0

.

A quadratic equation has exactly one real solution if and only if its discriminant is zero, meaning the parabola is tangent to the x-axis.

2

Identify the coefficients of the given quadratic equation

x^2 - px + q = 0

and substitute them into the discriminant formula.

Here,

a = 1

,

b = -p

, and

c = q

. Thus,

\Delta = (-p)^2 - 4(1)(q) = p^2 - 4q = 0

.

To apply the one-solution condition specifically to the coefficients of the given equation.

3

Solve the equation

p^2 - 4q = 0

for the ratio

\frac{p^2}{q}

.

p^2 = 4q \implies \frac{p^2}{q} = 4

.

Isolate the target expression by dividing both sides of the equation by

q

, which is valid since

q

is a positive constant and thus non-zero.

Key Concept

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

Estimated Time:1m 30s

Question 44Question →

In the quadratic equation $x^2 + kx + 36 = 0$ , $k$ is a constant. If the equation has two distinct real solutions, which of the following could be the value of $k$ ?

-12

-15

-10

8

Show answer & explanation

Answer: $-15$

Answer

The value

-15

is a possible value for

k

because it is the only option that satisfies the inequality

k < -12

or

k > 12

, which guarantees a positive discriminant and thus two distinct real solutions.

A quadratic equation has two distinct real solutions when its discriminant is positive. For

x^2 + kx + 36 = 0

, the discriminant is

k^2 - 4(1)(36) = k^2 - 144

. The inequality

k^2 - 144 > 0

simplifies to

k^2 > 144

, which means

k > 12

or

k < -12

. Among the choices, the value

-15

is the only one that satisfies this condition.

Step-by-Step Solution

1

Identify the condition for a quadratic equation to have two distinct real solutions.

The discriminant,

b^2 - 4ac

, must be strictly greater than

0

.

A positive discriminant ensures the quadratic formula yields two distinct real solutions.

2

Substitute the coefficients of the given equation

x^2 + kx + 36 = 0

into the discriminant formula.

Here,

a = 1

,

b = k

, and

c = 36

. The discriminant is

k^2 - 4(1)(36) = k^2 - 144

.

This expresses the discriminant in terms of the constant

k

.

3

Solve the inequality

k^2 - 144 > 0

for

k

.

k^2 > 144 \Rightarrow |k| > 12

, which means

k > 12

or

k < -12

.

This gives the range of values for

k

that produce two distinct real solutions.

4

Compare the given choices to the solution range.

Only

-15

satisfies the condition because

-15 < -12

.

This determines which of the options is a valid value for

k

.

Key Concept

Determining the number of real solutions of a quadratic equation using the discriminant.

Alternative Method

Instead of solving the inequality algebraically, substitute each answer choice for

k

into the equation and compute the discriminant

b^2 - 4ac

directly to identify which option yields a positive value.

Estimated Time:1m 30s

Question 45Question →

In the quadratic equation $3x^2 - 15x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $2$ , what is the value of the other solution?

3

-3

-7

5

Show answer & explanation

Answer: 3

Answer

The other solution to the equation is

3

.

Substituting

x = 2

into the quadratic equation gives

3(2)^2 - 15(2) + c = 0

, which simplifies to

12 - 30 + c = 0

, or

c = 18

. Substituting

c = 18

back into the original equation yields

3x^2 - 15x + 18 = 0

. Dividing the entire equation by

3

results in

x^2 - 5x + 6 = 0

. Factoring this equation gives

(x - 2)(x - 3) = 0

, which means the solutions are

x = 2

and

x = 3

. Therefore, the other solution is

3

. Alternatively, the sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-b/a

. For this equation, the sum of the roots is

-(-15)/3 = 5

. Since one root is

2

, the other root must be

5 - 2 = 3

.

Step-by-Step Solution

1

Substitute the given solution

x = 2

into the quadratic equation to find the value of the constant

c

.

3(2)^2 - 15(2) + c = 0 \implies 12 - 30 + c = 0 \implies c = 18

Since

2

is a solution, it must satisfy the equation when substituted for

x

.

2

Substitute

c = 18

back into the original quadratic equation and simplify by dividing both sides by

3

.

3x^2 - 15x + 18 = 0 \implies x^2 - 5x + 6 = 0

Dividing the equation by the greatest common factor simplifies the expression and makes it easier to factor.

3

Factor the simplified quadratic equation to find both solutions.

(x - 2)(x - 3) = 0 \implies x = 2

or

x = 3

The solutions to the factored equation are the values of

x

that make each factor equal to zero.

Key Concept

Solving quadratic equations by substitution and factoring

Estimated Time:1m 30s

Question 46Question →

In the equation $x(x - 8) = k$ , $k$ is a constant. If the product of the two real solutions to the equation is $-20$ , what is the value of the larger solution?

Show answer & explanation

Answer: 10

Answer

The larger solution to the equation is 10.

To find the larger solution, the equation is first rewritten in standard form as

x^2 - 8x - k = 0

. The product of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is

\frac{c}{a}

. Here,

a = 1

and

c = -k

, so the product of the roots is

-k

. Given that the product of the roots is

-20

, we can set up the equation

-k = -20

, which gives

k = 20

. Substituting

k = 20

back into the equation yields

x^2 - 8x - 20 = 0

. Factoring the quadratic expression gives

(x - 10)(x + 2) = 0

, which has the solutions

x = 10

and

x = -2

. The larger of these two solutions is 10.

Step-by-Step Solution

1

Rewrite the given equation

x(x - 8) = k

in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x - k = 0

To apply quadratic properties, the equation must be in standard form.

2

Use the product of roots formula to find the value of the constant

k

.

-k = -20

, which simplifies to

k = 20

The product of the roots of a quadratic equation in standard form is the constant term divided by the leading coefficient.

3

Substitute the value of

k

back into the quadratic equation and factor it to find the two solutions.

(x - 10)(x + 2) = 0

, so

x = 10

or

x = -2

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

4

Compare the two solutions to identify the larger value.

10

Comparing 10 and -2, 10 is the greater value.

Key Concept

Using the relationship between coefficients and the product of roots to solve a quadratic equation.

Question 47Question →

The quadratic equation $x^2 - 12x + 4 = 0$ has solutions $x_1$ and $x_2$ . What is the value of $\frac{1}{x_1} + \frac{1}{x_2}$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By writing the expression

\frac{1}{x_1} + \frac{1}{x_2}

with a common denominator, we get

\frac{x_1 + x_2}{x_1 x_2}

. For the quadratic equation

x^2 - 12x + 4 = 0

, Vieta's formulas give the sum of the roots

x_1 + x_2 = 12

and the product of the roots

x_1 x_2 = 4

. Substituting these values into the fraction yields

\frac{12}{4} = 3

.

Step-by-Step Solution

1

Find a common denominator to combine the terms in the given expression.

\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2}

To express the target quantity in terms of the sum and product of the quadratic solutions.

2

Apply Vieta's formulas to find the sum and product of the solutions from the quadratic equation

x^2 - 12x + 4 = 0

.

x_1 + x_2 = 12

and

x_1 x_2 = 4

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

3

Substitute the sum and product values into the combined fraction.

\frac{12}{4} = 3

To compute the numerical value of the expression.

Key Concept

Sum and product of solutions of a quadratic equation

Question 48Question →

In the equation $(2x - 3)(x + 4) = k$ , $k$ is a constant. If $x = 2$ is a solution to the equation, what is the other solution to the equation?

-\frac{9}{2}

\frac{9}{2}

\frac{1}{2}

-9

Show answer & explanation

Answer: $-\frac{9}{2}$

Answer

-\frac{9}{2}

Substituting the known solution

x = 2

into the equation

(2x - 3)(x + 4) = k

yields

(2(2) - 3)(2 + 4) = k

, which simplifies to

(1)(6) = k

, so

k = 6

. Substituting this value back into the equation gives

(2x - 3)(x + 4) = 6

. Expanding the left side yields

2x^2 + 5x - 12 = 6

. Subtracting 6 from both sides places the quadratic equation in standard form:

2x^2 + 5x - 18 = 0

. Since

x = 2

is a root,

(x - 2)

is a factor. Factoring the quadratic gives

(x - 2)(2x + 9) = 0

. Setting the second factor equal to zero,

2x + 9 = 0

, yields the other solution,

x = -\frac{9}{2}

.

Step-by-Step Solution

1

Substitute the known solution

x = 2

into the equation to solve for

k

.

k = 6

Since

x = 2

is a solution, it must satisfy the equation

(2(2) - 3)(2 + 4) = k

, which simplifies to

(1)(6) = k

.

2

Substitute

k = 6

back into the original equation, expand the binomial product, and write the quadratic equation in standard form.

2x^2 + 5x - 18 = 0

Expanding the binomials gives

2x^2 + 5x - 12 = 6

. Subtracting 6 from both sides yields the standard form quadratic equation

2x^2 + 5x - 18 = 0

.

3

Factor the quadratic equation to find the other root.

(x - 2)(2x + 9) = 0

, yielding

x = 2

and

x = -\frac{9}{2}

Since

x = 2

is a solution,

(x - 2)

must be a factor. Dividing the quadratic by

(x - 2)

yields the other factor,

(2x + 9)

.

Key Concept

Solving quadratic equations by substituting a known root to determine constants, then rewriting and factoring the equation.

Alternative Method

Another way to find the other solution is to use the relationship between the coefficients of a quadratic equation and its roots. Once the equation is written in standard form as

2x^2 + 5x - 18 = 0

, the sum of the roots is given by

-\frac{b}{a} = -\frac{5}{2}

. Since one root is

2

, the other root

r

must satisfy

2 + r = -\frac{5}{2}

, which simplifies to

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

.

Estimated Time:1m 30s

Question 49Question →

In the quadratic equation $x^2 - 10x + c = 0$ , $c$ is a constant. If the two real solutions to the equation have a difference of 6, what is the value of $c$ ?

Show answer & explanation

Answer: 16

Answer

16

The correct answer is 16. By using the quadratic formula, the two solutions of the equation

x^2 - 10x + c = 0

are

5 + \sqrt{25 - c}

and

5 - \sqrt{25 - c}

. The difference between these two solutions is

2\sqrt{25 - c}

. Given that the difference is 6, we set

2\sqrt{25 - c} = 6

, which simplifies to

\sqrt{25 - c} = 3

. Squaring both sides gives

25 - c = 9

, which yields

c = 16

.

Step-by-Step Solution

1

Use the quadratic formula to express the solutions of

x^2 - 10x + c = 0

.

The solutions are

x = 5 \pm \sqrt{25 - c}

.

This expresses the roots of the quadratic equation in terms of the constant

c

.

2

Set the difference between the two solutions equal to 6.

(5 + \sqrt{25 - c}) - (5 - \sqrt{25 - c}) = 6

, which simplifies to

2\sqrt{25 - c} = 6

.

We are given that the two real solutions have a difference of 6.

3

Solve the equation

2\sqrt{25 - c} = 6

for

c

.

\sqrt{25 - c} = 3 \implies 25 - c = 9 \implies c = 16

.

This isolates the constant

c

using standard algebraic operations.

Key Concept

Solving quadratic equations and using properties of roots.

Alternative Method

Alternatively, we can use the relationship between the roots of a quadratic equation. If the roots are

x_1

and

x_2

, then

x_1 + x_2 = 10

and

x_1 x_2 = c

. Using the identity

(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 x_2

, we substitute the given values:

(6)^2 = (10)^2 - 4c

. This simplifies to

36 = 100 - 4c

, which gives

4c = 64

, or

c = 16

.

Estimated Time:1m 30s

Question 50Question →

The quadratic equation $x^2 - 8x - 9 = 0$ can be written in the equivalent form $(x - a)^2 - b = 0$ , where $a$ and $b$ are positive constants. What is the value of $a + b$ ?

11

13

29

21

Show answer & explanation

Answer: 29

Answer

The value of

a + b

is 29.

Completing the square on the quadratic equation

x^2 - 8x - 9 = 0

yields

(x - 4)^2 - 25 = 0

. Comparing this to the form

(x - a)^2 - b = 0

shows that

a = 4

and

b = 25

. The sum of these values is

4 + 25 = 29

.

Step-by-Step Solution

1

Identify the coefficient of the linear term and find half of its value.

The coefficient of the linear term

-8x

is

-8

. Half of

-8

is

-4

.

This is the first step in completing the square.

2

Square the value obtained in the first step and add/subtract it in the equation to form a perfect square trinomial.

(-4)^2 = 16

. The equation becomes

(x^2 - 8x + 16) - 16 - 9 = 0

.

Adding and subtracting

16

maintains the equality while allowing us to group the first three terms as a perfect square.

3

Rewrite the perfect square trinomial and combine the remaining constant terms.

(x - 4)^2 - 25 = 0

.

This simplifies the equation into the desired equivalent form

(x - a)^2 - b = 0

.

4

Compare the equation to the target form

(x - a)^2 - b = 0

to identify the constants

a

and

b

, and calculate

a + b

.

a = 4

and

b = 25

. Therefore,

a + b = 4 + 25 = 29

.

This answers the question by finding the sum of the positive constants

a

and

b

.

Key Concept

Completing the square to rewrite a quadratic equation

Question 51Question →

In the equation $x^2 - bx + 16 = 0$ , $b$ is a positive integer. If the equation has no real solutions, what is the greatest possible value of $b$ ?

Show answer & explanation

Answer: 7

Answer

The greatest possible value of

b

is

7

.

For the quadratic equation

x^2 - bx + 16 = 0

to have no real solutions, its discriminant must be less than

0

. The discriminant is

(-b)^2 - 4(1)(16) = b^2 - 64

. Solving the inequality

b^2 - 64 < 0

gives

b^2 < 64

. Since

b

is a positive integer, taking the square root of both sides gives

b < 8

. The positive integers less than

8

are

1, 2, 3, 4, 5, 6,

and

7

. The greatest of these values is

7

.

Step-by-Step Solution

1

Set up the inequality for the discriminant to be less than zero.

(-b)^2 - 4(1)(16) < 0

A quadratic equation of the form

ax^2 + bx + c = 0

has no real solutions if and only if its discriminant,

D = b^2 - 4ac

, is negative.

2

Simplify the quadratic inequality.

b^2 < 64

Squaring

-b

yields

b^2

and calculating

4(1)(16)

yields

64

.

3

Solve for the greatest positive integer value of

b

.

b < 8

, so the greatest positive integer is

7

.

Since

b

is a positive integer, the values satisfying

b^2 < 64

are

1, 2, 3, 4, 5, 6,

and

7

. The largest of these is

7

.

Key Concept

Quadratic Discriminant and Number of Solutions

Question 52Question →

A retail store models its weekly profit, $P(d)$ , in dollars, from selling a certain product at a discount of $d$ dollars using the function $P(d) = -5(d - 8)^2 + 2,500$ , where $0 \leq d \leq 20$ . Which of the following is the best interpretation of the value 8 in this context?

The maximum weekly profit, in dollars, that the store can achieve.

The discount, in dollars, that results in the maximum weekly profit.

The weekly profit, in dollars, when no discount is offered.

The discount, in dollars, that results in the minimum weekly profit.

Show answer & explanation

Answer: The discount, in dollars, that results in the maximum weekly profit.

Answer

The discount, in dollars, that results in the maximum weekly profit.

The quadratic function

P(d) = -5(d - 8)^2 + 2,500

is given in vertex form,

P(d) = a(d - h)^2 + k

, where

(h, k)

is the vertex of the parabola. In this equation, the vertex is

(8, 2500)

. The leading coefficient is

-5

, which is negative, meaning the parabola opens downward and the vertex represents the maximum point of the function. In this context,

d

is the discount in dollars and

P(d)

is the weekly profit. Therefore, the value 8 represents the discount of 8 dollars that results in the maximum weekly profit.

Step-by-Step Solution

1

Identify the form of the quadratic function.

The function

P(d) = -5(d - 8)^2 + 2,500

is in vertex form,

P(d) = a(d - h)^2 + k

, where

(h, k)

represents the vertex of the parabola.

Recognizing the vertex form allows us to directly identify the vertex coordinates without expanding the equation.

2

Determine the vertex and the direction the parabola opens.

The vertex is

(8, 2,500)

. Since the leading coefficient

a = -5

is negative, the parabola opens downward, meaning the vertex represents a maximum point.

The sign of the leading coefficient determines whether the vertex represents a maximum or a minimum.

3

Interpret the coordinates of the vertex in context.

The variable

d

is the discount in dollars, and

P(d)

is the weekly profit in dollars. Thus, at the vertex, the discount is 8 dollars, which corresponds to the maximum profit of 2,500 dollars.

Matching the coordinates of the vertex to the variables in the context gives the correct real-world meaning.

Key Concept

Interpreting the vertex of a quadratic function in vertex form within a real-world context.

Estimated Time:1m 0s

Question 53Question →

In the quadratic equation $x^2 - 6x - k = 0$ , $k$ is a positive constant. If the solutions to the equation are $x = 3 \pm \sqrt{17}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

By applying the quadratic formula to

x^2 - 6x - k = 0

, we find the solutions are

x = 3 \pm \sqrt{9 + k}

. Equating the expression inside the radical to the given solutions

3 \pm \sqrt{17}

yields

9 + k = 17

. Solving for

k

gives 8.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = 1

,

b = -6

, and

c = -k

To apply the quadratic formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Substitute the coefficients into the quadratic formula.

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-k)}}{2(1)}

The quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

gives the solutions directly.

3

Simplify the expression inside the radical and divide by 2.

x = 3 \pm \sqrt{9 + k}

Simplifying the radical expression allows us to easily compare it to the given solutions.

4

Compare the simplified solution to the given solution form.

9 + k = 17

Since the solutions are given as

3 \pm \sqrt{17}

, the expression inside the radical must equal 17.

5

Solve for the constant

k

.

k = 8

Subtracting 9 from both sides of the equation yields the value of the constant.

Key Concept

Solving quadratic equations using the quadratic formula or by completing the square to find the relationship between coefficients and solutions.

Alternative Method

Instead of using the quadratic formula, the equation can be solved by completing the square. Rewrite the equation as

x^2 - 6x = k

. Adding 9 to both sides gives

x^2 - 6x + 9 = k + 9

, which can be factored as

(x - 3)^2 = k + 9

. Taking the square root of both sides yields

x = 3 \pm \sqrt{k + 9}

. Comparing this to the given solutions

3 \pm \sqrt{17}

, we get

k + 9 = 17

, so

k = 8

.

Estimated Time:1m 30s

Question 54Question →

If $(x - 2)^2 - 9 = 0$ and $x < 0$ , what is the value of $x^2 + 4x$ ?

5

-3

-5

45

Show answer & explanation

Answer: -3

Answer

The value of the expression is -3

To find the value of

x^2 + 4x

, we first solve the equation

(x - 2)^2 - 9 = 0

for

x

. Adding 9 to both sides gives

(x - 2)^2 = 9

. Taking the square root of both sides yields

x - 2 = 3

or

x - 2 = -3

. Solving these two linear equations gives

x = 5

or

x = -1

. The problem states that

x < 0

, so we must choose

x = -1

. Substituting

x = -1

into the expression

x^2 + 4x

gives

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

.

Step-by-Step Solution

1

Isolate the squared term in the equation.

(x - 2)^2 = 9

To solve a quadratic equation of the form

(x - h)^2 - k = 0

, we first add

k

to both sides to isolate the squared binomial.

2

Take the square root of both sides of the equation and solve for

x

.

x - 2 = \pm 3

, which gives

x = 5

or

x = -1

.

Taking the square root of both sides introduces both positive and negative roots, allowing us to find all possible values of

x

.

3

Apply the given constraint

x < 0

to determine the correct value of

x

.

x = -1

The problem specifies that

x

must be less than 0, so the positive solution

x = 5

must be rejected.

4

Substitute the value of

x

into the expression

x^2 + 4x

and simplify.

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

We substitute

x = -1

into the expression to find its final numerical value.

Key Concept

Solving quadratic equations by taking square roots and evaluating expressions under constraints

Estimated Time:1m 30s

Question 55Question →

If $(2y - 3)^2 - 2(2y - 3) - 24 = 0$ and $y > 0$ , what is the value of $y$ ?

\frac{9}{2}

-\frac{1}{2}

\frac{7}{2}

6

Show answer & explanation

Answer: $\frac{9}{2}$

Answer

\frac{9}{2}

Substituting

u = 2y - 3

transforms the original equation into the standard quadratic form

u^2 - 2u - 24 = 0

. Factoring this equation yields

(u - 6)(u + 4) = 0

, giving the solutions

u = 6

and

u = -4

. Substituting

2y - 3

back for

u

results in two equations:

2y - 3 = 6

(which simplifies to

y = \frac{9}{2}

) and

2y - 3 = -4

(which simplifies to

y = -\frac{1}{2}

). Since the problem specifies that

y > 0

, the negative value must be discarded, leaving the correct value as

\frac{9}{2}

.

Step-by-Step Solution

1

Use substitution to simplify the equation by letting

u = 2y - 3

.

u^2 - 2u - 24 = 0

This substitution reduces the equation to a standard quadratic form, making it easier to factor.

2

Factor the quadratic equation

u^2 - 2u - 24 = 0

.

(u - 6)(u + 4) = 0

, which gives

u = 6

or

u = -4

.

Factoring helps identify the potential values for the substituted expression

u

.

3

Substitute

2y - 3

back for

u

and solve both resulting linear equations for

y

.

2y - 3 = 6 \implies 2y = 9 \implies y = \frac{9}{2}

and

2y - 3 = -4 \implies 2y = -1 \implies y = -\frac{1}{2}

.

This step converts the solutions for the intermediate variable

u

back into solutions for the original variable

y

.

4

Apply the given constraint

y > 0

to identify the final solution.

y = \frac{9}{2}

because

-\frac{1}{2}

is not greater than

0

.

The question specifies that

y

must be strictly positive, so we must discard any negative values.

Key Concept

Solving quadratic equations using substitution and factoring under constraints

Question 56Question →

(2x - 5)(2x + 7) = -k

In the given equation, $k$ is a constant. If the equation has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

36

Expanding the equation

(2x - 5)(2x + 7) = -k

gives

4x^2 + 4x - 35 = -k

. Adding

k

to both sides puts it in standard form

4x^2 + 4x + (k - 35) = 0

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

b^2 - 4ac = 4^2 - 4(4)(k - 35) = 0

. Simplifying this yields

16 - 16(k - 35) = 0

, which gives

1 - (k - 35) = 0

, so

k - 35 = 1

, and thus

k = 36

.

Step-by-Step Solution

1

Expand the left side of the equation and write it in standard form.

4x^2 + 4x + (k - 35) = 0

To apply the quadratic discriminant, the equation must be in the form

ax^2 + bx + c = 0

.

2

Identify the coefficients and set the discriminant

b^2 - 4ac

equal to 0.

4^2 - 4(4)(k - 35) = 0

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

3

Solve the resulting linear equation for the constant

k

.

k = 36

Isolating

k

gives the value that satisfies the condition of having exactly one real solution.

Key Concept

Quadratic Discriminant

Question 57Question →

If the quadratic equation $x^2 - bx + 16 = 0$ , where $b$ is a positive constant, has two real solutions such that one solution is $4$ times the other, what is the value of $b$ ?

Show answer & explanation

Answer: 10

Answer

The value of

b

is

10

.

By writing the roots as

r

and

4r

, the quadratic equation can be represented as

(x-r)(x-4r) = x^2 - 5rx + 4r^2 = 0

. Comparing this to the given equation

x^2 - bx + 16 = 0

, we establish that

4r^2 = 16

and

b = 5r

. Solving for

r

gives

r^2 = 4

, which means

r = \pm 2

. Since

b

is a positive constant, we select

r = 2

, yielding

b = 5(2) = 10

. Alternatively, using Vieta's formulas, the product of the roots is

r \cdot 4r = 16 \implies 4r^2 = 16 \implies r = \pm 2

, and the sum of the roots is

r + 4r = b \implies 5r = b

. Since

b > 0

, we find

b = 10

.

Step-by-Step Solution

1

Define the roots in terms of a single variable and express the quadratic equation in factored form.

Let the roots be

r

and

4r

. The factored form is

(x - r)(x - 4r) = 0

.

This allows us to relate the given relationship between the roots to the coefficients of the quadratic equation.

2

Expand the factored quadratic expression.

x^2 - 5rx + 4r^2 = 0

This transforms the equation into the standard form

x^2 + Bx + C = 0

so we can match coefficients.

3

Compare the expanded equation with the given equation

x^2 - bx + 16 = 0

.

4r^2 = 16

and

b = 5r

Matching corresponding coefficients allows us to solve for the unknown variables.

4

Solve the constant term equation for

r

.

r = 2

or

r = -2

Finding the value of

r

is necessary to calculate the value of

b

.

5

Calculate the value of

b

using the constraint that

b

is positive.

b = 5(2) = 10

Using the negative root

r = -2

would yield a negative value for

b

(

b = -10

), which violates the constraint that

b

must be a positive constant.

Key Concept

Relating the roots of a quadratic equation to its coefficients

Estimated Time:1m 30s

Question 58Question →

One of the solutions to the quadratic equation $x^2 + 8x + c = 0$ , where $c$ is a constant, is $x = -4 + \sqrt{5}$ . What is the value of $c$ ?

-11

11

21

59

Show answer & explanation

Answer: 11

Answer

11

The value of 11 is correct because completing the square of the equation

x^2 + 8x + c = 0

yields

(x + 4)^2 = 16 - c

, which gives solutions of the form

x = -4 \pm \sqrt{16 - c}

. Comparing this to the given solution

x = -4 + \sqrt{5}

shows that

16 - c = 5

, which simplifies to

c = 11

. Direct substitution of the solution back into the equation also confirms this result.

Step-by-Step Solution

1

Express the quadratic equation in vertex form by completing the square.

x^2 + 8x + c = (x + 4)^2 - 16 + c = 0

Completing the square allows us to easily solve for the variable in terms of the constant.

2

Isolate the squared term and solve for the variable.

(x + 4)^2 = 16 - c \implies x = -4 \pm \sqrt{16 - c}

Taking the square root of both sides gives the general solutions of the quadratic equation.

3

Equate the general solution to the given solution to find the value of the constant.

Since one solution is

-4 + \sqrt{5}

, we set

16 - c = 5

, which gives

c = 11

.

Matching the terms under the radical determines the value of the constant.

Key Concept

Solving quadratic equations by completing the square or using the quadratic formula

Question 59Question →

The function $f$ is defined by $f(x) = 2x^2 - 12x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a vertex at $(h, 5)$ , where $h$ is a constant. What is the value of $c$ ?

Show answer & explanation

Answer: 23

Answer

23

For a quadratic function in standard form

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

, the x-coordinate of the vertex is given by

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

. In this function,

a = 2

and

b = -12

, so

h = -\frac{-12}{2(2)} = 3

. The vertex coordinates are

(3, 5)

, which means

f(3) = 5

. Substituting

x = 3

into the function gives

2(3)^2 - 12(3) + c = 5

. Simplifying the terms gives

18 - 36 + c = 5

, which becomes

-18 + c = 5

. Adding

18

to both sides yields

c = 23

.

Step-by-Step Solution

1

Identify the x-coordinate formula for the vertex of a quadratic function

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

.

The x-coordinate

h

is given by

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

.

To find the axis of symmetry and the horizontal position of the vertex.

2

Substitute

a = 2

and

b = -12

from the given equation

f(x) = 2x^2 - 12x + c

into the vertex formula.

h = -\frac{-12}{2(2)} = 3

To calculate the specific x-coordinate of the vertex for this function.

3

Substitute the vertex coordinates

(3, 5)

into the function

f(x)

.

f(3) = 2(3)^2 - 12(3) + c = 5

Since the vertex lies on the graph of the function, its coordinates must satisfy the function's equation.

4

Simplify the equation and solve for the constant

c

.

18 - 36 + c = 5 \implies -18 + c = 5 \implies c = 23

To determine the final value of the constant

c

.

Key Concept

Vertex of a quadratic function

Question 60Question →

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the difference between the two solutions to the equation is $5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 13

Answer

The value of the positive constant

k

is

13

.

For the quadratic equation

x^2 - kx + 36 = 0

with solutions

r_1

and

r_2

, the sum of the solutions is

r_1 + r_2 = k

and the product of the solutions is

r_1 \cdot r_2 = 36

. Given that the difference between the two solutions is

5

, we can write

|r_1 - r_2| = 5

. Squaring both sides yields

(r_1 - r_2)^2 = 25

. Using the algebraic identity

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

, we substitute the known values to obtain

25 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is positive,

k = 13

. Alternatively, we can find two numbers whose product is

36

and whose difference is

5

. These numbers are

9

and

4

, because

9 \cdot 4 = 36

and

9 - 4 = 5

. The sum of these solutions is

9 + 4 = 13

, which matches the coefficient of the linear term,

k

.

Step-by-Step Solution

1

Relate the roots of the quadratic equation

x^2 - kx + 36 = 0

to its coefficients using Vieta's formulas.

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 \cdot r_2 = 36

.

This sets up the system of equations representing the roots.

2

Express the given root difference of

5

mathematically and square it.

|r_1 - r_2| = 5 \implies (r_1 - r_2)^2 = 25

.

Squaring the difference allows us to use standard algebraic identities.

3

Apply the identity

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

to substitute the sum and product expressions.

25 = k^2 - 4(36)

This converts the relationship between the roots into an equation with the single variable

k

.

4

Solve the equation for

k

, selecting the positive value.

25 = k^2 - 144 \implies k^2 = 169 \implies k = 13

(since

k > 0

).

To determine the final value of

k

satisfying the constraint that

k

is a positive constant.

Key Concept

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

Quadratic Equations

Answer

Step-by-Step Solution

Key Concept

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Key Concept

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Alternative Method

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Alternative Method

Answer

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Alternative Method

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Answer

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Key Concept

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Alternative Method

Answer

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Answer

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Common Mistakes

Answer

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Key Concept