Quadratic Equations and the Quadratic Formula Practice Questions — ACT — Page 2

Question 21Question →

The daily revenue $R$ , in dollars, of a manufacturing company is modeled by the quadratic function $R(x) = -0.2x^2 + kx - 1,200$ , where $x$ is the number of units produced and sold, and $k$ is a positive constant. If the maximum daily revenue the company can achieve is $800$ dollars, what is the value of $k$ ?

Show answer & explanation

Answer: 40

Answer

The value of the constant

k

is 40.

Setting the daily revenue function equal to 800 and rewriting it in standard form yields

-0.2x^2 + kx - 2000 = 0

. For a quadratic equation to have exactly one real solution, which represents the maximum vertex of the parabola, the discriminant must be equal to 0. Setting the discriminant

b^2 - 4ac = 0

gives

k^2 - 4(-0.2)(-2000) = 0

, which simplifies to

k^2 - 1600 = 0

. Solving for the positive constant

k

gives

k = 40

.

Step-by-Step Solution

1

Set the revenue function equal to the maximum daily revenue of 800 dollars.

-0.2x^2 + kx - 1,200 = 800

The maximum revenue is the highest point (vertex) on the parabola, where the line

y = 800

is tangent to the curve.

2

Convert the equation into standard quadratic form

ax^2 + bx + c = 0

.

-0.2x^2 + kx - 2,000 = 0

Standard form is required to identify the coefficients

a

,

b

, and

c

for the discriminant formula.

3

Set the discriminant equal to zero.

k^2 - 4(-0.2)(-2,000) = 0

Since the maximum daily revenue is achieved at exactly one point, the quadratic equation must have exactly one real solution, meaning its discriminant (

b^2 - 4ac

) must be zero.

4

Solve for the positive constant

k

.

k^2 - 1,600 = 0 \implies k = 40

Solving the equation yields

k = \pm 40

. Since the problem specifies that

k

is a positive constant, we select

k = 40

.

Key Concept

Using the discriminant of a quadratic equation to find the value of a parameter when there is exactly one real solution.

Estimated Time:2m 0s

Question 22Question →

One of the solutions to the quadratic equation $2x^2 - kx + 18 = 0$ is exactly four times the other solution. If $k$ is a positive constant, what is the value of $k$ ?

Show answer & explanation

Answer: 15

Answer

15

By representing the roots as

r

and

4r

, we can use Vieta's formulas to find that the product of the roots is

4r^2 = 9

, which gives

r = 1.5

. The sum of the roots is

5r = k/2

, which gives

k = 10(1.5) = 15

.

Step-by-Step Solution

1

Represent the roots of the quadratic equation.

Let the two solutions be

r

and

4r

, where

r

is a real number.

We are given that one solution is exactly four times the other.

2

Apply Vieta's formulas for the product of the roots.

r \cdot 4r = 4r^2 = \frac{18}{2} = 9

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is equal to

\frac{c}{a}

.

3

Solve for the root

r

.

r^2 = \frac{9}{4} \Rightarrow r = 1.5

(since

k

is positive,

r

must be positive)

Solving the equation

4r^2 = 9

gives

r = \pm 1.5

. Since the sum of the roots is positive, we select the positive root.

4

Apply Vieta's formulas for the sum of the roots to find

k

.

r + 4r = 5r = -\frac{-k}{2} = \frac{k}{2} \Rightarrow k = 10r = 10(1.5) = 15

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is equal to

-\frac{b}{a}

.

Key Concept

Using Vieta's formulas to relate the roots of a quadratic equation to its coefficients.

Question 23Question →

A rectangular painting is $2$ feet wide and $6$ feet long. A wooden frame of uniform width $x$ feet is placed around the painting. If the total area of the painting and the frame is $21$ square feet, what is the width of the frame, in feet?

1.5

1.0

0.5

4.5

8.0

Show answer & explanation

Answer: 0.5

Answer

The correct width of the frame is 0.5 feet.

The correct answer is 0.5 feet. To find the width of the frame, we define the total dimensions of the framed painting as

(2 + 2x)

and

(6 + 2x)

. Setting their product equal to the total area of 21 square feet gives the quadratic equation

4x^2 + 16x - 9 = 0

. Solving this using the quadratic formula yields

x = 0.5

and

x = -4.5

. Since a physical width must be positive, the width of the frame is 0.5 feet.

Step-by-Step Solution

1

Set up the equation for the total area. The painting's dimensions are

2

feet by

6

feet. Adding a frame of uniform width

x

on all sides increases both the width and the length by

2x

.

The total dimensions are

(2 + 2x)

and

(6 + 2x)

, and the total area is given by the equation:

(2 + 2x)(6 + 2x) = 21

.

To find the width of the frame, we must relate the final total area to the dimensions of the painting plus the frame.

2

Expand the equation and write it in standard quadratic form

ax^2 + bx + c = 0

.

12 + 4x + 12x + 4x^2 = 21 \implies 4x^2 + 16x + 12 = 21 \implies 4x^2 + 16x - 9 = 0

.

Standard form is required to apply the quadratic formula or to factor the quadratic expression.

3

Solve the quadratic equation

4x^2 + 16x - 9 = 0

using the quadratic formula:

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

.

x = \frac{-16 \pm \sqrt{16^2 - 4(4)(-9)}}{2(4)} = \frac{-16 \pm \sqrt{256 + 144}}{8} = \frac{-16 \pm \sqrt{400}}{8} = \frac{-16 \pm 20}{8}

.

This formula provides the solutions to any quadratic equation.

4

Calculate the two possible values for

x

and select the physically meaningful one.

x = \frac{-16 + 20}{8} = 0.5

or

x = \frac{-16 - 20}{8} = -4.5

. Since width must be positive, the only valid solution is

x = 0.5

.

A physical measurement like width cannot be negative.

Key Concept

Quadratic Equations and the Quadratic Formula

Question 24Question →

For the quadratic equation $2x^2 + bx + 16 = 0$ , where $b$ is a positive constant, the sum of the squares of the two complex solutions is equal to $-7$ . What is the value of $b$ ?

Show answer & explanation

Answer: 6

Answer

The value of the positive constant

b

is

6

.

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

2x^2 + bx + 16 = 0

is

-\frac{b}{2}

and the product is

8

. Using the identity

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

, we substitute

-7

for the sum of the squares, yielding

-7 = \left(-\frac{b}{2}\right)^2 - 2(8)

. This simplifies to

-7 = \frac{b^2}{4} - 16

, which leads to

\frac{b^2}{4} = 9

and

b^2 = 36

. Since

b

is a positive constant,

b = 6

.

Step-by-Step Solution

1

Find the sum and product of the roots in terms of

b

using Vieta's formulas.

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

and

x_1 x_2 = 8

Vieta's formulas state that for a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

2

Relate the sum of the squares of the roots to their sum and product.

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

This algebraic identity allows us to express the sum of the squares of the roots using the sum and product of the roots.

3

Substitute the known values into the identity and solve for

b

.

-7 = \left(-\frac{b}{2}\right)^2 - 2(8) \implies -7 = \frac{b^2}{4} - 16 \implies \frac{b^2}{4} = 9 \implies b^2 = 36 \implies b = 6

Substituting the given sum of squares (

-7

), sum (

-\frac{b}{2}

), and product (

8

) produces a single-variable equation that can be solved for the positive constant

b

.

Key Concept

Using Vieta's formulas and algebraic identities to relate the roots of a quadratic equation to its coefficients.

Question 25Question →

If $k$ is a non-zero constant, for what value of $k$ does the quadratic equation $(x - 3)^2 = kx$ have exactly one real solution?

6

-3

-12

-6

12

Show answer & explanation

Answer: $-12$

Answer

-12

Expanding

(x - 3)^2

yields

x^2 - 6x + 9 = kx

. Subtracting

kx

from both sides and grouping like terms gives

x^2 - (6 + k)x + 9 = 0

. For a quadratic equation to have exactly one real solution, its discriminant must be equal to zero. Thus, we set

(-(6+k))^2 - 4(1)(9) = 0

, which simplifies to

(6+k)^2 - 36 = 0

. Taking the square root of both sides gives

6 + k = 6

or

6 + k = -6

. Solving these equations gives

k = 0

or

k = -12

. Since the problem specifies that

k

is a non-zero constant, the correct value is

-12

.

Step-by-Step Solution

1

Expand the squared binomial on the left side of the equation.

x^2 - 6x + 9 = kx

Expanding the binomial

(x - 3)^2

allows us to rewrite the equation in a form where we can group terms.

2

Move

kx

to the left side and group the

x

terms to write the equation in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - (6 + k)x + 9 = 0

, where

a = 1

,

b = -(6 + k)

, and

c = 9

.

The coefficients

a

,

b

, and

c

must be identified from the standard quadratic form to calculate the discriminant.

3

Set the discriminant

b^2 - 4ac

equal to

0

and solve for

k

.

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

. This yields

k = 0

or

k = -12

. Since

k

is non-zero,

k = -12

.

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

Key Concept

Using the discriminant (

b^2 - 4ac = 0

) to determine when a quadratic equation has exactly one real solution.

Alternative Method

Alternatively, one can recognize that the equation

(x-3)^2 = kx

can be written as

x^2 - (6+k)x + 9 = 0

. For a quadratic equation with a leading coefficient of

1

and a constant term of

9

to have exactly one real solution, it must be a perfect square trinomial. A perfect square trinomial of the form

x^2 + bx + 9

must have

b = \pm 6

. Setting the middle coefficient equal to these values gives

-(6+k) = 6 \implies k = -12

or

-(6+k) = -6 \implies k = 0

. Since

k

is non-zero,

k = -12

.

Estimated Time:1m 30s

Question 26Question →

The quadratic equation $0.4x^2 - 2x + c = 0$ has two real solutions. If the difference between these two solutions is exactly $3$ , what is the value of $c$ ?

Show answer & explanation

Answer: 1.6

Answer

1.6

By using the relation between the roots of a quadratic equation

ax^2 + bx + c = 0

, we find that the difference of the roots is given by

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting

a = 0.4

,

b = -2

, and the difference of

3

yields

3 = \frac{\sqrt{4 - 1.6c}}{0.4}

. Multiplying by

0.4

and squaring both sides gives

1.44 = 4 - 1.6c

. Solving this linear equation gives

c = 1.6

. We can verify this result by substituting

c = 1.6

back into the original equation:

0.4x^2 - 2x + 1.6 = 0

simplifies to

x^2 - 5x + 4 = 0

, which factors as

(x-1)(x-4) = 0

. The roots are

1

and

4

, and their difference is

4 - 1 = 3

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = 0.4

,

b = -2

, and the constant term is

c

.

To apply formulas relating the roots to the coefficients of the equation.

2

Apply the formula for the difference of the roots.

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

The problem states the difference between the two solutions is

3

.

3

Substitute the known values into the formula and solve.

3 = \frac{\sqrt{(-2)^2 - 4(0.4)c}}{0.4} \Rightarrow 1.2 = \sqrt{4 - 1.6c}

To isolate the square root expression containing the unknown variable.

4

Square both sides and solve the linear equation for

c

.

1.44 = 4 - 1.6c \Rightarrow 1.6c = 2.56 \Rightarrow c = 1.6

To eliminate the square root and find the value of

c

.

Key Concept

Quadratic Equations and the Quadratic Formula

Estimated Time:1m 30s

Question 27Question →

For the quadratic equation $1.5x^2 - 5x + c = 0$ , where $c$ is a real constant, the equation has two non-real complex solutions. Which of the following inequalities represents all possible values of $c$ ?

c > \frac{25}{6}

c < \frac{25}{6}

c > -\frac{25}{6}

c > \frac{50}{3}

c < -\frac{25}{6}

Show answer & explanation

Answer: $c > \frac{25}{6}$

Answer

c > \frac{25}{6}

For a quadratic equation to have two non-real complex solutions, its discriminant must be negative. Substituting

a = 1.5

,

b = -5

, and the constant

c

into the discriminant formula

b^2 - 4ac < 0

gives

25 - 6c < 0

. Solving this inequality results in the requirement that the constant must be strictly greater than twenty-five sixths.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

1.5x^2 - 5x + c = 0

in the standard form

ax^2 + bx + c = 0

.

a = 1.5

,

b = -5

, and

c = c

.

These coefficients are required to compute the discriminant.

2

Set up the condition for the quadratic equation to have two non-real complex solutions using the discriminant

\Delta = b^2 - 4ac

.

The discriminant must be strictly negative:

(-5)^2 - 4(1.5)(c) < 0

.

A quadratic equation has non-real complex solutions if and only if its discriminant is negative.

3

Simplify the inequality and solve for

c

.

25 - 6c < 0 \implies 25 < 6c \implies c > \frac{25}{6}

.

Isolating

c

yields the range of values that satisfy the condition.

Key Concept

A quadratic equation

ax^2 + bx + c = 0

has two non-real complex solutions if and only if its discriminant,

\Delta = b^2 - 4ac

, is strictly less than zero.

Estimated Time:1m 30s

Question 28Question →

One of the solutions to the quadratic equation $0.5x^2 + bx - 6 = 0$ , where $b$ is a constant, is $x = 3$ . What is the value of the other solution?

Show answer & explanation

Answer: -4

Answer

The other solution to the quadratic equation is -4.

Substituting the given solution

x = 3

into the equation yields

0.5(3)^2 + 3b - 6 = 0

. Simplifying this expression gives

4.5 + 3b - 6 = 0

, which leads to

3b = 1.5

and thus

b = 0.5

. With

b = 0.5

, the quadratic equation becomes

0.5x^2 + 0.5x - 6 = 0

. Multiplying the entire equation by 2 to obtain integer coefficients results in

x^2 + x - 12 = 0

. This quadratic factors into

(x - 3)(x + 4) = 0

, which gives the solutions

x = 3

and

x = -4

. Therefore, the other solution is

-4

. Alternatively, using Vieta's formulas, the product of the roots of a quadratic equation

ax^2 + bx + c = 0

is equal to

c/a

. Here, the product of the roots is

-6 / 0.5 = -12

. Since one root is

3

, the other root must be

-12 / 3 = -4

.

Step-by-Step Solution

1

Substitute the given solution

x = 3

into the quadratic equation to find the value of

b

.

b = 0.5

Since

x = 3

is a solution, it must satisfy the equation, allowing us to solve for the unknown coefficient

b

.

2

Rewrite the equation using

b = 0.5

and simplify by multiplying all terms by 2.

x^2 + x - 12 = 0

Multiplying the equation by 2 eliminates the decimal coefficients, making the quadratic expression easier to factor.

3

Factor the quadratic equation to determine the roots.

x = 3

or

x = -4

The equation factors into

(x - 3)(x + 4) = 0

. Solving for

x

yields the given root of 3 and the second root of -4.

Key Concept

Solving quadratic equations by utilizing a known solution to determine unknown coefficients, and applying factoring techniques or root relationships to find the remaining solution.

Estimated Time:1m 30s

Question 29Question →

For what value of the constant $c$ does the quadratic equation $x^2 - 3x + c = 0$ have two complex solutions with imaginary parts equal to $\pm 2i$ ?

-1.75

1.75

2.25

3.25

6.25

Show answer & explanation

Answer: $6.25$

Answer

The value of the constant

c

is

6.25

.

The correct answer is

6.25

. Applying the quadratic formula to

x^2 - 3x + c = 0

gives solutions of the form

1.5 \pm \frac{\sqrt{9 - 4c}}{2}

. Since these solutions are complex with imaginary parts equal to

\pm 2i

, the term under the radical must be negative, and the imaginary component is

\frac{\sqrt{9 - 4c}}{2} = 2i

. Multiplying both sides by

2

gives

\sqrt{9 - 4c} = 4i

. Squaring both sides results in

9 - 4c = 16i^2

. Substituting

i^2 = -1

gives

9 - 4c = -16

. Solving for

c

yields

-4c = -25

, which simplifies to

c = 6.25

.

Step-by-Step Solution

1

Apply the quadratic formula to the equation

x^2 - 3x + c = 0

.

The solutions are given by

x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(c)}}{2(1)} = 1.5 \pm \frac{\sqrt{9 - 4c}}{2}

.

This expresses the solutions in terms of the constant

c

so that the imaginary part can be identified.

2

Set the imaginary term of the solutions equal to the given imaginary parts

\pm 2i

.

\frac{\sqrt{9 - 4c}}{2} = 2i \implies \sqrt{9 - 4c} = 4i

.

The question specifies that the imaginary parts of the two complex solutions are

\pm 2i

.

3

Square both sides of the equation to solve for

c

.

9 - 4c = (4i)^2 = 16i^2

. Since

i^2 = -1

, this becomes

9 - 4c = -16

.

Squaring eliminates the radical and allows for standard algebraic isolation of the variable

c

.

4

Solve the linear equation for

c

.

-4c = -25 \implies c = 6.25

.

Subtracting

9

from both sides and then dividing by

-4

isolates the constant

c

.

Key Concept

Solving quadratic equations with complex roots using the quadratic formula and the properties of the imaginary unit.

Question 30Question →

In the quadratic equation $2x^2 - 11x + c = 0$ , where $c$ is a constant, the ratio of the two real solutions is $3:8$ . What is the value of $c$ ?

Show answer & explanation

Answer: 12

Answer

The value of the constant

c

is

12

.

The correct answer is

12

. By representing the roots in the ratio of

3:8

as

3r

and

8r

, Vieta's formula for the sum of roots (

-\frac{b}{a}

) gives

3r + 8r = -\frac{-11}{2} \implies 11r = 5.5 \implies r = 0.5

. The actual roots are therefore

1.5

and

4

. Using Vieta's formula for the product of roots (

\frac{c}{a}

) gives

(1.5)(4) = \frac{c}{2} \implies 6 = \frac{c}{2} \implies c = 12

.

Step-by-Step Solution

1

Represent the roots using the given ratio.

Let the two roots of the quadratic equation be

3r

and

8r

.

The ratio of the two solutions is specified as

3:8

.

2

Apply Vieta's formula for the sum of roots to find the ratio multiplier

r

.

3r + 8r = -\frac{-11}{2} \implies 11r = 5.5 \implies r = 0.5

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is given by

-\frac{b}{a}

.

3

Determine the numerical values of the two roots.

The roots are

3(0.5) = 1.5

and

8(0.5) = 4

.

Substitute the value of

r = 0.5

back into the expressions for the roots.

4

Apply Vieta's formula for the product of roots to solve for the constant

c

.

(1.5)(4) = \frac{c}{2} \implies 6 = \frac{c}{2} \implies c = 12

.

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is given by

\frac{c}{a}

.

Key Concept

Vieta's formulas and the relationship between the roots and coefficients of a quadratic equation

Alternative Method

Alternatively, you can express the roots using the quadratic formula:

x = \frac{11 \pm \sqrt{121 - 8c}}{4}

. Since the ratio of the smaller root to the larger root is

3:8

, we set up the equation:

\frac{11 - \sqrt{121 - 8c}}{11 + \sqrt{121 - 8c}} = \frac{3}{8}

. Cross-multiplying gives

8(11 - \sqrt{121 - 8c}) = 3(11 + \sqrt{121 - 8c}) \implies 88 - 8\sqrt{121 - 8c} = 33 + 3\sqrt{121 - 8c} \implies 55 = 11\sqrt{121 - 8c} \implies 5 = \sqrt{121 - 8c} \implies 25 = 121 - 8c \implies 8c = 96 \implies c = 12

.

Estimated Time:1m 30s

Question 31Question →

A right triangle has a hypotenuse of length $x + 4$ inches. The lengths of the two legs of the triangle are $x$ inches and $x + 2$ inches. What is the value of $x$ ?

2

6

8

10

12

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6 because applying the Pythagorean theorem yields the relation

x^2 + (x + 2)^2 = (x + 4)^2

. Expanding the binomials gives

x^2 + x^2 + 4x + 4 = x^2 + 8x + 16

, which simplifies to the quadratic equation

x^2 - 4x - 12 = 0

. Factoring this equation yields

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

. Discarding the negative solution

x = -2

because length must be positive leaves the correct solution of 6.

Step-by-Step Solution

1

Set up the equation using the Pythagorean theorem, where the sum of the squares of the legs equals the square of the hypotenuse.

x^2 + (x + 2)^2 = (x + 4)^2

The sides of a right triangle must satisfy the Pythagorean relation

a^2 + b^2 = c^2

.

2

Expand the squared binomial terms on both sides of the equation.

x^2 + (x^2 + 4x + 4) = x^2 + 8x + 16

Applying the algebraic identity

(a+b)^2 = a^2 + 2ab + b^2

is necessary to simplify the terms.

3

Simplify the equation and move all terms to one side to set the quadratic expression to zero.

x^2 - 4x - 12 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve quadratic equations.

4

Factor the quadratic equation.

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

Factoring allows finding the roots by setting each linear binomial factor to zero.

5

Solve for

x

and discard any physically impossible negative values.

x = 6

(since

x = -2

is discarded)

A physical measurement of side length must be strictly positive.

Key Concept

Formulating and solving quadratic equations derived from the Pythagorean theorem by expanding binomials and factoring.

Question 32Question →

A toy rocket is launched upward from a platform. Its height $h$ , in meters, above the ground $t$ seconds after launch is modeled by the function $h(t) = -4.9t^2 + 7.35t + 12.25$ . According to this model, how many seconds after launch does the rocket strike the ground?

Show answer & explanation

Answer: 2.5

Answer

The rocket strikes the ground

2.5

seconds after launch.

The correct answer of

2.5

is determined by setting the height

h(t)

to

0

and solving the resulting quadratic equation using the quadratic formula. Since time must be non-negative in this physical context, the negative solution of

-1

is discarded, leaving

2.5

seconds as the time when the rocket strikes the ground.

Step-by-Step Solution

1

Set the height function

h(t)

to

0

.

-4.9t^2 + 7.35t + 12.25 = 0

The rocket strikes the ground when its height above the ground is

0

meters.

2

Apply the quadratic formula

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

.

t = \frac{-7.35 \pm \sqrt{(7.35)^2 - 4(-4.9)(12.25)}}{2(-4.9)}

This formula provides the solutions to any quadratic equation of the form

at^2 + bt + c = 0

.

3

Calculate the discriminant and its square root.

b^2 - 4ac = 294.1225

and

\sqrt{294.1225} = 17.15

Evaluating the term under the radical simplifies the quadratic formula expression.

4

Evaluate the two possible values for

t

.

t = -1

or

t = 2.5

Solving the simplified expression gives the two mathematical roots of the quadratic equation.

5

Choose the physically valid solution.

t = 2.5

Time must be positive in this scenario, so the negative solution

t = -1

is discarded.

Key Concept

Solving a quadratic equation with decimal coefficients using the quadratic formula in a real-world motion context.

Question 33Question →

A quadratic equation is defined by $x^2 - bx + 18 = 0$ , where $b$ is a positive constant. If the difference between the two real solutions of this equation is 3, what is the value of $b$ ?

Show answer & explanation

Answer: 9

Answer

The value of the positive constant

b

is 9.

By applying the quadratic formula, the roots of the equation are

x = \frac{b \pm \sqrt{b^2 - 72}}{2}

. The difference between these roots is

\sqrt{b^2 - 72}

. Setting this equal to 3 gives

\sqrt{b^2 - 72} = 3

. Squaring both sides yields

b^2 - 72 = 9

, which simplifies to

b^2 = 81

. Taking the positive root since

b

is a positive constant gives

b = 9

.

Step-by-Step Solution

1

Express the roots of the quadratic equation

x^2 - bx + 18 = 0

using the quadratic formula.

The roots are

x = \frac{b \pm \sqrt{b^2 - 4(1)(18)}}{2} = \frac{b \pm \sqrt{b^2 - 72}}{2}

.

This provides a formulaic representation of the two solutions in terms of the unknown parameter

b

.

2

Subtract the smaller root from the larger root to represent the difference between the solutions, and set this expression equal to 3.

Difference

= \frac{b + \sqrt{b^2 - 72}}{2} - \frac{b - \sqrt{b^2 - 72}}{2} = \sqrt{b^2 - 72} = 3

.

The problem specifies that the difference between the two real solutions is 3.

3

Square both sides of the equation to eliminate the radical, and solve for the positive constant

b

.

b^2 - 72 = 9 \Rightarrow b^2 = 81 \Rightarrow b = 9

(since

b

is positive).

Squaring both sides allows us to isolate

b^2

and find the value of

b

that satisfies the initial condition.

Key Concept

Solving for quadratic coefficients using the difference of roots derived from the quadratic formula.

Alternative Method

Use Vieta's formulas. Let the roots be

r_1

and

r_2

. We know that

r_1 + r_2 = b

and

r_1 r_2 = 18

. We are given that the difference between the roots is 3, so

|r_1 - r_2| = 3

. We can use the algebraic identity

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

. Substituting the known values gives

3^2 = b^2 - 4(18)

, which simplifies to

9 = b^2 - 72

, leading to

b^2 = 81

. Since

b > 0

, we find

b = 9

.

Estimated Time:1m 30s

Question 34Question →

A model glider is launched from a hill. Its height $h(t)$ , in meters above the valley floor $t$ seconds after launch, is modeled by the function $h(t) = -0.5t^2 + 3.5t + 10$ . Which of the following is a possible value of $t$ , in seconds, when the glider is at a height of exactly $12$ meters?

\frac{7 + \sqrt{33}}{2}

\frac{7 + \sqrt{65}}{2}

\frac{-7 + \sqrt{33}}{2}

7 + \sqrt{33}

11

Show answer & explanation

Answer: $\frac{7 + \sqrt{33}}{2}$

Answer

\frac{7 + \sqrt{33}}{2}

The correct answer is obtained by setting the height equation

h(t) = 12

, which simplifies to

-0.5t^2 + 3.5t - 2 = 0

. Multiplying by

-2

gives the standard form

t^2 - 7t + 4 = 0

. Applying the quadratic formula with

a = 1

,

b = -7

, and

c = 4

gives

t = \frac{7 \pm \sqrt{33}}{2}

. Thus, the option representing

\frac{7 + \sqrt{33}}{2}

is the correct choice.

Step-by-Step Solution

1

Set the height function equal to the target height of 12 meters.

-0.5t^2 + 3.5t + 10 = 12

To find when the glider reaches exactly 12 meters, we set the model function equal to 12.

2

Subtract 12 from both sides to set the quadratic equation to zero.

-0.5t^2 + 3.5t - 2 = 0

A quadratic equation must be in standard form

at^2 + bt + c = 0

before applying the quadratic formula.

3

Multiply the entire equation by

-2

to eliminate decimal coefficients.

t^2 - 7t + 4 = 0

Working with integer coefficients reduces calculation errors when applying the quadratic formula. Here,

a = 1

,

b = -7

, and

c = 4

.

4

Apply the quadratic formula

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

.

t = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(4)}}{2(1)} = \frac{7 \pm \sqrt{49 - 16}}{2} = \frac{7 \pm \sqrt{33}}{2}

The quadratic formula is used to solve quadratic equations that cannot be easily factored using integers.

Key Concept

Solving quadratic equations with decimal coefficients by converting to standard integer form and applying the quadratic formula.

Question 35Question →

For what value of $c$ does the quadratic equation $0.5x^2 - 3x + c = 0$ have two real solutions that differ by exactly 4?

-2.5

2.5

4

5

10

Show answer & explanation

Answer: 2.5

Answer

2.5

The correct value is 2.5. By utilizing the formula for the difference of the roots,

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

, and substituting

a = 0.5

and

b = -3

, we get

\frac{\sqrt{9 - 2c}}{0.5} = 4

. This simplifies to

\sqrt{9 - 2c} = 2

, which squares to

9 - 2c = 4

. Solving for

c

yields 2.5.

Step-by-Step Solution

1

Identify the coefficients and apply the relationship for the difference between two roots.

For a quadratic equation

ax^2 + bx + c = 0

, the roots

x_1

and

x_2

satisfy

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Here,

a = 0.5

,

b = -3

, and the difference is 4.

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

2

Substitute the given values into the formula and solve for

c

.

Substituting the values gives

\frac{\sqrt{(-3)^2 - 4(0.5)c}}{|0.5|} = 4 \Rightarrow \frac{\sqrt{9 - 2c}}{0.5} = 4

. Multiplying both sides by 0.5 yields

\sqrt{9 - 2c} = 2

. Squaring both sides gives

9 - 2c = 4

.

Simplifying the equation isolates the variable

c

under the radical.

3

Complete the algebraic isolation to find the final value of

c

.

2c = 5 \Rightarrow c = 2.5

.

This final step solves the linear equation for

c

.

Key Concept

Using the discriminant and properties of roots to solve quadratic equations with given constraints.

Question 36Question →

For the quadratic equation $0.25x^2 - 1.5x + c = 0$ , the discriminant is equal to $4$ . What is the value of the constant $c$ ?

-7.00

-6.25

-1.75

1.75

6.25

Show answer & explanation

Answer: $-1.75$

Answer

The value of the constant

c

is

-1.75

.

The correct option is

-1.75

. To find the value of

c

, substitute the coefficients

a = 0.25

,

b = -1.5

, and the discriminant

D = 4

into the formula

D = b^2 - 4ac

. This gives

4 = (-1.5)^2 - 4(0.25)c

, which simplifies to

4 = 2.25 - c

. Solving for

c

yields

c = 2.25 - 4 = -1.75

.

Step-by-Step Solution

1

Identify the values of the coefficients from the quadratic equation

0.25x^2 - 1.5x + c = 0

.

The coefficients are

a = 0.25

,

b = -1.5

, and the constant is

c

.

These coefficients are required to compute the discriminant.

2

Recall the formula for the discriminant

D

and substitute the known values, including the given discriminant

D = 4

.

4 = (-1.5)^2 - 4(0.25)c

This sets up an equation to solve for the unknown constant

c

.

3

Simplify the squared term and the multiplication of the coefficients.

4 = 2.25 - 1c

, which simplifies to

4 = 2.25 - c

.

Squaring

-1.5

yields

2.25

, and

4(0.25) = 1

.

4

Solve the linear equation for

c

.

c = -1.75

Subtracting

2.25

from both sides gives

1.75 = -c

, which means

c = -1.75

.

Key Concept

Using the discriminant formula

D = b^2 - 4ac

to solve for an unknown coefficient in a quadratic equation.

Estimated Time:1m 30s

Question 37Question →

The quadratic equation $1.5x^2 - kx + 6 = 0$ , where $k$ is a positive constant, has exactly one real solution. What is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

6

For the quadratic equation

1.5x^2 - kx + 6 = 0

to have exactly one real solution, the discriminant

b^2 - 4ac

must equal

0

. Substituting

a = 1.5

,

b = -k

, and

c = 6

gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

, which yields

k^2 = 36

. Since

k

must be a positive constant,

k

must be

6

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

1.5x^2 - kx + 6 = 0

.

a = 1.5

,

b = -k

, and

c = 6

To use the discriminant formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Set the discriminant equal to zero.

b^2 - 4ac = 0

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

3

Substitute the coefficients into the discriminant formula and simplify.

k^2 - 36 = 0

Substituting

a = 1.5

,

b = -k

, and

c = 6

into the formula gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

.

4

Solve the equation for the positive constant

k

.

k = 6

Solving

k^2 = 36

gives

k = 6

or

k = -6

. Since the problem states that

k

is a positive constant, we choose

k = 6

.

Key Concept

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

Question 38Question →

A right triangle has legs of length $x - 0.5$ inches and $2x$ inches, and a hypotenuse of length $2x + 0.5$ inches. What is the value of $x$ ?

1

1.5

2

3

4

Show answer & explanation

Answer: 3

Answer

3

Applying the Pythagorean theorem to the right triangle yields

(x - 0.5)^2 + (2x)^2 = (2x + 0.5)^2

. Expanding the terms gives

x^2 - x + 0.25 + 4x^2 = 4x^2 + 2x + 0.25

. Subtracting

4x^2

and

0.25

from both sides simplifies the equation to

x^2 - x = 2x

. Subtracting

2x

from both sides gives the standard quadratic equation

x^2 - 3x = 0

. Factoring this expression gives

x(x - 3) = 0

, which yields solutions

x = 0

and

x = 3

. Since the side length

x - 0.5

must be positive,

x

must be greater than

0.5

. Therefore, the only valid solution is

3

.

Step-by-Step Solution

1

Set up the equation using the Pythagorean theorem,

a^2 + b^2 = c^2

, with the given side lengths.

(x - 0.5)^2 + (2x)^2 = (2x + 0.5)^2

The Pythagorean theorem relates the legs and hypotenuse of any right triangle.

2

Expand each squared term algebraically.

(x^2 - x + 0.25) + 4x^2 = 4x^2 + 2x + 0.25

Expanding the binomials allows us to combine like terms and simplify the equation.

3

Subtract

4x^2

and

0.25

from both sides of the equation.

x^2 - x = 2x

Simplifying the equation makes it easier to solve.

4

Move all terms to the left side to write the quadratic equation in standard form.

x^2 - 3x = 0

A quadratic equation must be set to zero to be solved by factoring.

5

Factor the quadratic expression.

x(x - 3) = 0

Factoring allows us to find the roots of the equation.

6

Solve for

x

and choose the value that makes all side lengths positive.

x = 3

(since

x = 0

is not a valid length because a side length

x - 0.5

must be greater than

0

)

Only a positive value of

x

greater than

0.5

yields physically possible side lengths for the triangle.

Key Concept

Setting up and solving quadratic equations using algebraic expansion and the Pythagorean theorem

Estimated Time:1m 30s

Question 39Question →

A projectile is launched vertically upward from an initial height of $5$ meters. Its height, $h(t)$ in meters, $t$ seconds after launch is given by the function $h(t) = -4.9t^2 + 19.6t + 5$ . To the nearest tenth of a second, how many seconds after launch does the projectile reach a height of $15$ meters on its way down?

Show answer & explanation

Answer: 3.4

Answer

To the nearest tenth of a second, the projectile reaches a height of

15

meters on its way down at

3.4

seconds.

The correct answer is

3.4

seconds. Setting the height equation

h(t) = 15

yields

-4.9t^2 + 19.6t - 10 = 0

. Solving this quadratic equation via the quadratic formula gives two solutions:

t \approx 0.6

seconds and

t \approx 3.4

seconds. The projectile travels upward first, passing the

15

-meter mark at

0.6

seconds, and then descends, passing the

15

-meter mark again at

3.4

seconds.

Step-by-Step Solution

1

Set up the quadratic equation by setting the height function

h(t)

equal to

15

.

-4.9t^2 + 19.6t + 5 = 15

This allows us to find the specific values of time

t

when the height of the projectile is exactly

15

meters.

2

Rearrange the quadratic equation into the standard form

at^2 + bt + c = 0

by subtracting

15

from both sides.

-4.9t^2 + 19.6t - 10 = 0

Writing the equation in standard form is necessary before applying the quadratic formula.

3

Substitute the coefficients

a = -4.9

,

b = 19.6

, and

c = -10

into the quadratic formula.

t = \frac{-19.6 \pm \sqrt{(19.6)^2 - 4(-4.9)(-10)}}{2(-4.9)}

Since the quadratic equation has non-integer decimal coefficients, using the quadratic formula is the most reliable method to solve for the roots.

4

Simplify the discriminant and calculate the two values of

t

.

t \approx 0.6

and

t \approx 3.4

The discriminant is

19.6^2 - 196 = 188.16

. Taking the square root gives

\sqrt{188.16} \approx 13.72

, resulting in two real roots.

5

Determine which root corresponds to the projectile's motion on the way down.

t \approx 3.4

seconds

The smaller root (

0.6

seconds) represents the first time the projectile reaches

15

meters while ascending. The larger root (

3.4

seconds) represents the time the projectile passes

15

meters while descending.

Key Concept

Solving quadratic equations with decimal coefficients using the quadratic formula and interpreting the physical context of the roots.

Question 40Question →

What are the solutions for $x$ in the quadratic equation $0.2x^2 - 0.6x + 2.2 = 0$ ?

\frac{3 \pm \sqrt{35}}{2}

\frac{3 \pm \sqrt{55}}{2}

\frac{3 \pm i\sqrt{35}}{2}

\pm\frac{i\sqrt{35}}{2}

\frac{15 \pm 2i\sqrt{10}}{10}

Show answer & explanation

Answer: $\frac{3 \pm i\sqrt{35}}{2}$

Answer

The correct answer is

\frac{3 \pm i\sqrt{35}}{2}

.

The correct answer is the pair of complex solutions

\frac{3 \pm i\sqrt{35}}{2}

. By multiplying the equation

0.2x^2 - 0.6x + 2.2 = 0

by

5

, we get the equivalent equation with integer coefficients,

x^2 - 3x + 11 = 0

. Applying the quadratic formula with

a = 1

,

b = -3

, and

c = 11

yields a discriminant of

9 - 44 = -35

. Since the discriminant is negative, the solutions are complex:

\frac{3 \pm i\sqrt{35}}{2}

.

Step-by-Step Solution

1

Multiply both sides of the quadratic equation by

5

to eliminate the decimal coefficients.

The equation

0.2x^2 - 0.6x + 2.2 = 0

becomes

x^2 - 3x + 11 = 0

.

Working with integer coefficients simplifies algebraic manipulation and reduces the risk of calculation errors.

2

Identify the coefficients

a

,

b

, and

c

in the standard quadratic form

ax^2 + bx + c = 0

.

a = 1

,

b = -3

, and

c = 11

.

These values are required to apply the quadratic formula.

3

Substitute the coefficients into the quadratic formula,

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

.

x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(11)}}{2(1)} = \frac{3 \pm \sqrt{9 - 44}}{2} = \frac{3 \pm \sqrt{-35}}{2}

.

The quadratic formula provides the exact solutions for any quadratic equation.

4

Simplify the radical using the imaginary unit,

i = \sqrt{-1}

.

\sqrt{-35} = \sqrt{35} \cdot \sqrt{-1} = i\sqrt{35}

, so the solutions are

x = \frac{3 \pm i\sqrt{35}}{2}

.

Standard mathematical notation represents the square root of a negative number using

i

.

Key Concept

Using the quadratic formula to solve quadratic equations with decimal coefficients and complex roots.

Quadratic Equations and the Quadratic Formula

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Step-by-Step Solution

Key Concept

Common Mistakes