Quadratic Equations and the Quadratic Formula Practice Questions — ACT

What is the sum of the squares of the two real solutions to the quadratic equation $\frac{1}{2}x^2 - 3x + 2 = 0$ ?

Show answer & explanation

Answer: 28

Answer

The sum of the squares of the two real solutions is 28.

Multiplying the equation

\frac{1}{2}x^2 - 3x + 2 = 0

by 2 yields

x^2 - 6x + 4 = 0

. By Vieta's formulas, the sum of the solutions is

x_1 + x_2 = 6

and the product of the solutions is

x_1 x_2 = 4

. Using the algebraic identity

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

, the sum of the squares is

6^2 - 2(4) = 36 - 8 = 28

. Alternatively, using the quadratic formula on the simplified equation gives solutions of

3 + \sqrt{5}

and

3 - \sqrt{5}

. Squaring these values yields

14 + 6\sqrt{5}

and

14 - 6\sqrt{5}

, which sum to 28.

Step-by-Step Solution

1

Multiply the quadratic equation by 2 to clear the fraction.

x^2 - 6x + 4 = 0

Simplifying the fractional coefficients makes the equation easier to analyze and solve.

2

Identify the sum and product of the roots using Vieta's formulas.

x_1 + x_2 = 6

and

x_1 x_2 = 4

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-b/a

and the product is

c/a

.

3

Express the sum of the squares of the roots using the algebraic identity.

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

This identity allows the direct calculation of the sum of squares without needing to find the individual roots.

4

Substitute the values of the sum and product into the identity.

6^2 - 2(4) = 36 - 8 = 28

Plugging in the sum of 6 and product of 4 yields the final numerical answer.

Key Concept

Vieta's formulas state that for a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

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

and the product of the roots is

x_1 x_2 = \frac{c}{a}

. Symmetric functions of roots like

x_1^2 + x_2^2

can be expressed in terms of these values.

Alternative Method

Find the roots of the equation directly using the quadratic formula. After simplifying to

x^2 - 6x + 4 = 0

, the roots are

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(4)}}{2} = 3 \pm \sqrt{5}

. Squaring both solutions gives

(3 + \sqrt{5})^2 = 14 + 6\sqrt{5}

and

(3 - \sqrt{5})^2 = 14 - 6\sqrt{5}

. Adding these squares together yields

(14 + 6\sqrt{5}) + (14 - 6\sqrt{5}) = 28

.

Estimated Time:1m 30s

Question 2Question →

A quadratic equation of the form $a x^2 + b x + c = 0$ , where $a$ , $b$ , and $c$ are real constants and $a > 0$ , has a discriminant of $37$ . If the sum of the roots of this equation is $5.5$ and the product of the roots is $5.25$ , what is the value of the coefficient $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of the coefficient

a

is

2

.

By applying Vieta's formulas, we can write

b = -5.5a

and

c = 5.25a

. Plugging these into the discriminant formula gives

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

. Setting the discriminant to

37

results in

9.25a^2 = 37 \implies a^2 = 4

. Since the problem specifies

a > 0

, taking the positive square root gives

a = 2

.

Step-by-Step Solution

1

Express the coefficients

b

and

c

in terms of

a

using Vieta's formulas.

b = -5.5a

and

c = 5.25a

The sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

2

Substitute the expressions for

b

and

c

into the discriminant formula

D = b^2 - 4ac

.

D = 9.25a^2

Substituting the terms yields

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

.

3

Equate the discriminant expression to

37

and solve for

a

.

a = 2

Since

D = 37

, we write

9.25a^2 = 37 \implies a^2 = 4

. Because

a

must be positive, we find

a = 2

.

Key Concept

Using the properties of quadratic roots (Vieta's formulas) and the definition of the discriminant to solve for coefficients.

Question 3Question →

In the quadratic equation $a x^2 + b x + c = 0$ , the coefficients $a$ , $b$ , and $c$ are real numbers, and $a = \frac{2}{3}$ . If one of the roots of this equation is $x = \frac{1}{2} - i\sqrt{2}$ , where $i = \sqrt{-1}$ , what is the value of the product $bc$ ?

-1

-\frac{7}{9}

-\frac{1}{2}

0

\frac{7}{9}

Show answer & explanation

Answer: $-1$

Answer

-1

For any quadratic equation with real coefficients, if a complex number is a root, its conjugate must also be a root. Thus, the two roots are

x_1 = \frac{1}{2} - i\sqrt{2}

and

x_2 = \frac{1}{2} + i\sqrt{2}

. Using Vieta's formulas, the sum of the roots is

x_1 + x_2 = 1 = -\frac{b}{a}

, which gives

b = -\frac{2}{3}

. The product of the roots is

x_1 x_2 = \frac{1}{4} - 2i^2 = \frac{9}{4} = \frac{c}{a}

, which gives

c = \frac{3}{2}

. The product

bc

is therefore

(-\frac{2}{3})(\frac{3}{2}) = -1

.

Step-by-Step Solution

1

Find the second root of the quadratic equation.

The second root is

x_2 = \frac{1}{2} + i\sqrt{2}

.

Since the quadratic equation has real coefficients, complex roots must occur in conjugate pairs.

2

Calculate the sum of the roots to find the coefficient

b

.

b = -\frac{2}{3}

The sum of the roots is

x_1 + x_2 = (\frac{1}{2} - i\sqrt{2}) + (\frac{1}{2} + i\sqrt{2}) = 1

. By Vieta's formulas,

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

. Substituting

a = \frac{2}{3}

gives

1 = -\frac{b}{2/3} \implies b = -\frac{2}{3}

.

3

Calculate the product of the roots to find the coefficient

c

.

c = \frac{3}{2}

The product of the roots is

x_1 x_2 = (\frac{1}{2} - i\sqrt{2})(\frac{1}{2} + i\sqrt{2}) = (\frac{1}{2})^2 - (i\sqrt{2})^2 = \frac{1}{4} - 2i^2

. Since

i^2 = -1

, the product is \frac{1}{4} - 2(-1) = \frac{9}{4}

. By Vieta's formulas,

x_1 x_2 = \frac{c}{a}

. Substituting

a = \frac{2}{3}

gives \frac{9}{4} = \frac{c}{2/3} \implies c = \frac{3}{2}

.

4

Calculate the product

bc

.

bc = -1

Multiplying the values of

b

and

c

gives

bc = (-\frac{2}{3})(\frac{3}{2}) = -1

.

Key Concept

Relating the roots of a quadratic equation with real coefficients to its coefficients via Vieta's formulas and the Complex Conjugate Root Theorem.

Question 4Question →

A projectile is launched from ground level with an initial upward velocity of $v$ meters per second. The height of the projectile in meters after $t$ seconds is modeled by the equation $h(t) = -\frac{9}{2}t^2 + vt$ . A sensor is placed at a height of $32$ meters. What is the minimum integer value of $v$ for which the projectile is at or above the height of the sensor for a duration of at least $4$ seconds?

Show answer & explanation

Answer: 30

Answer

The minimum integer value of the initial upward velocity

v

is 30.

The height condition

h(t) \ge 32

leads to the quadratic inequality

9t^2 - 2vt + 64 \le 0

. The roots

t_1

and

t_2

of

9t^2 - 2vt + 64 = 0

represent the boundary times. The duration above

32

meters is the difference of the roots:

t_2 - t_1 = \frac{\sqrt{b^2 - 4ac}}{|a|} = \frac{\sqrt{4v^2 - 2304}}{9}

. Setting this to be at least

4

seconds gives

\frac{\sqrt{4v^2 - 2304}}{9} \ge 4

. Squaring both sides after multiplying by

9

yields

4v^2 - 2304 \ge 1296

, which simplifies to

4v^2 \ge 3600

, or

v^2 \ge 900

. Taking the positive square root gives

v \ge 30

, meaning the minimum integer value is

30

.

Step-by-Step Solution

1

Write the inequality representing the condition that the projectile is at or above the sensor height.

9t^2 - 2vt + 64 \le 0

The height function

h(t) = -\frac{9}{2}t^2 + vt

must be greater than or equal to

32

meters.

2

Use the formula for the difference of roots to represent the time duration.

t_2 - t_1 = \frac{\sqrt{4v^2 - 2304}}{9}

The duration of time the projectile is above the sensor is the difference between the two times at which the projectile crosses the sensor height.

3

Set the duration to be at least

4

seconds and solve for

v

.

v \ge 30

Solving the inequality

\frac{\sqrt{4v^2 - 2304}}{9} \ge 4

yields the range of initial velocities.

4

Find the smallest integer in the solution set.

30

Since the velocity must be an integer and at least

30

, the minimum possible integer value is

30

.

Key Concept

Solving quadratic inequalities and finding the difference of roots using the discriminant.

Question 5Question →

A rectangular photograph has a length of $x$ inches and a width that is $1.5$ inches shorter than its length. If the area of the photograph is $10$ square inches, what is the length, in inches, of the photograph?

2.5

3.25

4

5

5.75

Show answer & explanation

Answer: 4

Answer

4

The correct length is 4. When the length is 4 inches, the width is 1.5 inches shorter, which is 2.5 inches. The area is the product of these dimensions: 4 * 2.5 = 10 square inches, which matches the given condition.

Step-by-Step Solution

1

Set up the equation for the area of the photograph.

x(x - 1.5) = 10

The area of a rectangle is found by multiplying its length (

x

) by its width (

x - 1.5

).

2

Distribute the

x

and subtract

10

from both sides to write the equation in standard form.

x^2 - 1.5x - 10 = 0

Standard form for a quadratic equation is

ax^2 + bx + c = 0

.

3

Multiply the entire equation by

2

to clear the decimal coefficient.

2x^2 - 3x - 20 = 0

Working with integer coefficients makes factoring or applying the quadratic formula easier.

4

Factor the quadratic expression.

(2x + 5)(x - 4) = 0

Factoring helps find the roots of the equation directly.

5

Solve for

x

by setting each factor to zero.

x = -2.5

or

x = 4

If the product of two numbers is zero, at least one of the numbers must be zero.

6

Exclude the negative solution.

x = 4

The length of a photograph must be a positive value.

Key Concept

Solving a quadratic equation by translating a word problem and factoring.

Question 6Question →

For the quadratic equation $3x^2 - 6x + 2 = 0$ , what is the value of the discriminant?

Show answer & explanation

Answer: 12

Answer

The discriminant of the quadratic equation is 12.

The discriminant of a quadratic equation

ax^2 + bx + c = 0

is determined by the expression

b^2 - 4ac

. By substituting the coefficients

a = 3

,

b = -6

, and

c = 2

from the given equation

3x^2 - 6x + 2 = 0

, we calculate

(-6)^2 - 4(3)(2) = 36 - 24 = 12

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

a = 3

,

b = -6

, and

c = 2

The standard form of a quadratic equation is

ax^2 + bx + c = 0

.

2

Substitute the coefficients into the discriminant formula

D = (-6)^2 - 4(3)(2)

The discriminant formula is

D = b^2 - 4ac

.

3

Simplify the expression to find the final value

36 - 24 = 12

Squaring

-6

gives

36

and multiplying

4 \times 3 \times 2

gives

24

. Subtracting

24

from

36

gives

12

.

Key Concept

Calculating the discriminant of a quadratic equation

Estimated Time:45s

Question 7Question →

What are the solutions to the quadratic equation $x^2 - \frac{5}{2}x + 1 = 0$ ?

x = -\frac{1}{2}

and

x = -2

x = 1

and

x = \frac{1}{4}

x = \frac{1}{2}

and

x = 2

x = \frac{5}{4} + \frac{3}{4}i

and

x = \frac{5}{4} - \frac{3}{4}i

x = \frac{3}{4}

and

x = -\frac{3}{4}

Show answer & explanation

Answer: $x = \frac{1}{2}$ and $x = 2$

Answer

The solutions are

x = \frac{1}{2}

and

x = 2

.

The correct answer provides the exact solutions to the equation. Multiplying the equation by

2

yields

2x^2 - 5x + 2 = 0

. Factoring this expression gives

(2x - 1)(x - 2) = 0

, which yields the solutions

x = \frac{1}{2}

and

x = 2

.

Step-by-Step Solution

1

Multiply the entire equation by

2

to clear the fraction.

2x^2 - 5x + 2 = 0

Converting the equation to integer coefficients simplifies factoring or applying the quadratic formula.

2

Identify the coefficients

a

,

b

, and

c

to use in the quadratic formula.

a = 2

,

b = -5

,

c = 2

These coefficients correspond to the standard quadratic form

ax^2 + bx + c = 0

.

3

Calculate the discriminant

b^2 - 4ac

.

(-5)^2 - 4(2)(2) = 25 - 16 = 9

The discriminant determines the number and type of solutions.

4

Substitute the coefficients and the discriminant into the quadratic formula:

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

.

x = \frac{-(-5) \pm \sqrt{9}}{2(2)} = \frac{5 \pm 3}{4}

This formula provides the exact solutions to any quadratic equation.

5

Evaluate the two cases for the plus/minus sign.

x = \frac{5 + 3}{4} = 2

and

x = \frac{5 - 3}{4} = \frac{1}{2}

Splitting the formula into positive and negative branches yields the two distinct real roots.

Key Concept

Solving quadratic equations using the quadratic formula

Question 8Question →

A right triangle has an area of $6$ square units. The lengths of its legs, $a$ and $b$ , satisfy the quadratic equation $a^2 - 2.75ab + 1.5b^2 = 0$ . If the length of the hypotenuse is an integer, what is the perimeter of the triangle?

9

10

12

14

15

Show answer & explanation

Answer: 12

Answer

The perimeter of the triangle is 12.

The correct answer is 12. By solving the quadratic relationship between the legs, we find two possible ratios: one where one leg is twice the other, and one where one leg is 0.75 times the other. Using the area of 6, the first case yields non-integer side lengths, while the second case yields legs of length 3 and 4. This results in an integer hypotenuse of 5, giving a perimeter of 3 + 4 + 5 = 12.

Step-by-Step Solution

1

Express the relation

a^2 - 2.75ab + 1.5b^2 = 0

as a quadratic in terms of the ratio

r = \frac{a}{b}

.

The equation becomes

r^2 - 2.75r + 1.5 = 0

.

To find the relationship between the two legs of the triangle by solving for their ratio.

2

Apply the quadratic formula to solve for

r

.

r = 2

or

r = 0.75

, meaning either

a = 2b

or

a = 0.75b

.

To determine the two possible linear relationships between the legs of the right triangle.

3

Substitute each ratio case into the area formula

\text{Area} = \frac{1}{2}ab = 6

, which simplifies to

ab = 12

.

For

a = 2b

, we get

b = \sqrt{6}

and

a = 2\sqrt{6}

. For

a = 0.75b

, we get

b = 4

and

a = 3

.

To calculate the actual leg lengths for both geometric cases.

4

Calculate the hypotenuse

c = \sqrt{a^2 + b^2}

for both cases to check which yields an integer value.

The first case yields

c = \sqrt{30}

, which is not an integer. The second case yields

c = \sqrt{3^2 + 4^2} = 5

, which is an integer.

To satisfy the constraint that the hypotenuse must be an integer, identifying the correct leg lengths as

3

and

4

.

5

Calculate the perimeter of the triangle for the valid case.

The perimeter is

3 + 4 + 5 = 12

.

To find the final requested value.

Key Concept

Solving quadratic relationships and applying the quadratic formula in geometric constraints.

Estimated Time:3m 0s

Question 9Question →

What is the sum of the two solutions to the quadratic equation $2x^2 - 7x - 4 = 0$ ?

Show answer & explanation

Answer: 3.5

Answer

The sum of the solutions is

3.5

.

The sum of the solutions of the quadratic equation

2x^2 - 7x - 4 = 0

is

3.5

. According to Vieta's formulas, the sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

. Here,

a = 2

and

b = -7

, so the sum is

-\frac{-7}{2} = 3.5

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

4

and

-0.5

, which sum to

3.5

.

Step-by-Step Solution

1

Identify the coefficients from the quadratic equation

2x^2 - 7x - 4 = 0

.

a = 2

,

b = -7

, and

c = -4

.

A quadratic equation in standard form is written as

ax^2 + bx + c = 0

.

2

Apply the sum of roots formula

-\frac{b}{a}

.

Sum

= -\frac{-7}{2} = 3.5

.

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

ax^2 + bx + c = 0

is

-\frac{b}{a}

.

Key Concept

Sum of roots of a quadratic equation using Vieta's formulas

Alternative Method

Solve the quadratic equation by factoring or using the quadratic formula. Factoring

2x^2 - 7x - 4 = 0

yields

(2x + 1)(x - 4) = 0

, which gives solutions

x = -0.5

and

x = 4

. Adding these solutions together gives

-0.5 + 4 = 3.5

.

Estimated Time:1m 0s

Question 10Question →

One of the solutions to the quadratic equation $0.5x^2 - x + c = 0$ is $1 + i$ , where $i = \sqrt{-1}$ and $c$ is a constant. What is the value of $c$ ?

0

1 - 2i

1

-1

1 + i

Show answer & explanation

Answer: 1

Answer

The constant

c

is equal to

1

.

Substituting the given solution

x = 1 + i

into the equation

0.5x^2 - x + c = 0

gives

0.5(1+i)^2 - (1+i) + c = 0

. Since

(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

, the equation simplifies to

0.5(2i) - 1 - i + c = 0

, which is

i - 1 - i + c = 0

. Simplifying this yields

-1 + c = 0

, so the value of

c

is

1

.

Step-by-Step Solution

1

Substitute the solution

x = 1 + i

into the quadratic equation

0.5x^2 - x + c = 0

.

0.5(1+i)^2 - (1+i) + c = 0

Since

1 + i

is a solution to the equation, substituting it for

x

must satisfy the equation.

2

Expand the squared term

(1+i)^2

using the binomial square formula and the identity

i^2 = -1

.

(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

Expanding

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

with

a=1

and

b=i

, and replacing

i^2

with

-1

simplifies the term.

3

Substitute

2i

back into the equation and simplify.

0.5(2i) - (1+i) + c = 0 \Rightarrow i - 1 - i + c = 0 \Rightarrow -1 + c = 0

Multiplying

0.5

by

2i

yields

i

, distributing the negative sign across

(1+i)

yields

-1 - i

, and combining like terms leaves

-1 + c = 0

.

4

Solve the linear equation for

c

.

c = 1

Adding

1

to both sides isolates the variable

c

.

Key Concept

Solving quadratic equations with complex roots by substitution and algebraic manipulation.

Question 11Question →

If $3 - 2i$ is a root of the quadratic equation $x^2 + bx + c = 0$ , where $b$ and $c$ are real numbers and $i = \sqrt{-1}$ , what is the value of $b + c$ ?

-11

-5

-1

7

19

Show answer & explanation

Answer: 7

Answer

7

The correct answer is

7

. Since the quadratic equation has real coefficients, the roots must be complex conjugates. The conjugate of the root

3 - 2i

is

3 + 2i

. Vieta's formulas show that the sum of the roots is

-b

, meaning

(3 - 2i) + (3 + 2i) = 6 = -b

, which yields

b = -6

. The product of the roots is

c

, meaning

(3 - 2i)(3 + 2i) = 9 - 4i^2 = 9 - 4(-1) = 13

. Adding these coefficients together gives

b + c = -6 + 13 = 7

.

Step-by-Step Solution

1

Identify the second root of the quadratic equation

The second root is

3 + 2i

.

Since the quadratic equation has real coefficients, the Complex Conjugate Theorem dictates that if a complex number is a root, its complex conjugate must also be a root.

2

Determine the coefficient

b

using the sum of the roots

b = -6

According to Vieta's formulas, the sum of the roots of the equation

x^2 + bx + c = 0

is equal to

-b

. Thus,

(3 - 2i) + (3 + 2i) = 6 = -b

, which simplifies to

b = -6

.

3

Determine the constant term

c

using the product of the roots

c = 13

According to Vieta's formulas, the product of the roots of the equation

x^2 + bx + c = 0

is equal to

c

. Thus,

(3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2

. Substituting

i^2 = -1

yields

9 - 4(-1) = 9 + 4 = 13

.

4

Calculate the value of

b + c

b + c = 7

Substitute the calculated values of

b

and

c

into the expression:

-6 + 13 = 7

.

Key Concept

The Complex Conjugate Theorem states that complex roots of polynomials with real coefficients occur in conjugate pairs. Vieta's formulas state that for a quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

.

Question 12Question →

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

Show answer & explanation

Answer: 49

Answer

The discriminant of the quadratic equation is 49.

The discriminant is calculated using the formula

b^2 - 4ac

. For the equation

3x^2 + 5x - 2 = 0

, the coefficients are

a = 3

,

b = 5

, and

c = -2

. Substituting these yields

5^2 - 4(3)(-2) = 25 - (-24) = 25 + 24 = 49

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

3x^2 + 5x - 2 = 0

in the standard form

ax^2 + bx + c = 0

.

a = 3

,

b = 5

, and

c = -2

To use the discriminant formula, we must first extract the constant coefficients corresponding to each term.

2

Substitute the coefficients into the discriminant formula

D = b^2 - 4ac

.

D = 5^2 - 4(3)(-2) = 25 - (-24) = 25 + 24 = 49

Calculating the value of the discriminant provides the required solution.

Key Concept

Calculating the discriminant of a quadratic equation to determine the nature of its roots.

Question 13Question →

For a constant $k$ , the quadratic equation $\frac{1}{3}x^2 - \frac{k}{6}x + (k - 10) = 0$ has two real roots, $r_1$ and $r_2$ . If the sum of the reciprocals of the roots, $\frac{1}{r_1} + \frac{1}{r_2}$ , is equal to $-\frac{1}{2}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7.5

Answer

The value of

k

is

7.5

.

By writing the sum of the reciprocals in terms of the sum and product of the roots, we find that

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

. Substituting the values from Vieta's formulas (

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

and

r_1 r_2 = 3(k - 10)

) yields

\frac{k}{6(k - 10)} = -\frac{1}{2}

. Solving this equation for

k

results in

k = 7.5

.

Step-by-Step Solution

1

Identify the quadratic coefficients in terms of the constant

k

.

a = \frac{1}{3}

,

b = -\frac{k}{6}

, and

c = k - 10

Applying Vieta's formulas requires the coefficients from the standard form

ax^2 + bx + c = 0

.

2

Determine the sum and the product of the roots using Vieta's formulas.

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

and

r_1 r_2 = 3(k - 10)

Vieta's formulas state that the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

3

Express the sum of the reciprocals of the roots in terms of

k

.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = \frac{k}{6(k - 10)}

Finding a common denominator allows the sum of the reciprocals to be written as the ratio of the sum of the roots to the product of the roots.

4

Equate the expression to

-\frac{1}{2}

and solve the resulting equation for

k

.

k = 7.5

Solving

\frac{k}{6(k - 10)} = -\frac{1}{2}

gives

2k = -6(k - 10) \implies 8k = 60 \implies k = 7.5

.

Key Concept

Vieta's Formulas and Algebraic Relationships of Roots

Estimated Time:2m 30s

Question 14Question →

The length of a rectangle is $3$ inches greater than its width. If the area of the rectangle is $10$ square inches, what is the width of the rectangle, in inches?

1

2

3.5

5

7

Show answer & explanation

Answer: $2$

Answer

The width of the rectangle is

2

inches.

The correct answer is the option stating that the width is

2

inches. This is obtained by expressing the length as

w+3

and setting the area equation to

w(w+3) = 10

. Expanding and setting it to zero gives the quadratic equation

w^2 + 3w - 10 = 0

, which factors to

(w+5)(w-2) = 0

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

2

.

Step-by-Step Solution

1

Define variables for the dimensions of the rectangle based on the problem statement.

Let the width of the rectangle be

w

inches. The length of the rectangle is

w + 3

inches.

The problem states that the length is

3

inches greater than the width.

2

Set up an equation representing the area of the rectangle.

The area is the product of width and length:

w(w + 3) = 10

.

The area of a rectangle is calculated as width times length, and the given area is

10

square inches.

3

Rewrite the equation in standard quadratic form

aw^2 + bw + c = 0

.

Distribute

w

to get

w^2 + 3w = 10

, then subtract

10

from both sides:

w^2 + 3w - 10 = 0

.

Standard quadratic form is required to solve the equation by factoring or using the quadratic formula.

4

Factor the quadratic equation.

(w + 5)(w - 2) = 0

.

Finding factors that multiply to

-10

and add to

3

allows us to find the roots.

5

Solve for

w

and apply real-world constraints.

w = -5

or

w = 2

. Since a physical width must be positive, discard

-5

, leaving

w = 2

.

A dimension of a geometric shape cannot be negative.

Key Concept

Solving quadratic equations derived from geometric word problems by factoring or using the quadratic formula.

Question 15Question →

For a real number $a$ , the quadratic equation $x^2 + ax + (a - 3)^2 = 0$ has complex roots. One of these roots is $2 + i\sqrt{b}$ , where $b$ is a positive real number and $i = \sqrt{-1}$ . What is the value of $a + b$ ?

17

-21

41

-49

1

Show answer & explanation

Answer: 41

Answer

The value of

a + b

is

41

.

The correct answer is

41

. Since the quadratic equation has real coefficients, its complex roots must occur in conjugate pairs. Thus, the roots are

2 + i\sqrt{b}

and

2 - i\sqrt{b}

. Their sum is

4

, which by Vieta's formulas equals

-a

, giving

a = -4

. The product of the roots is

4 + b

, which equals the constant term

(a - 3)^2 = (-4 - 3)^2 = 49

. Solving for

b

gives

b = 45

. Therefore,

a + b = -4 + 45 = 41

.

Step-by-Step Solution

1

Identify the second root using the complex conjugate root theorem.

The conjugate root is

2 - i\sqrt{b}

.

Since the coefficients of the quadratic equation are real, complex roots must occur in conjugate pairs.

2

Find the value of

a

using the sum of the roots.

a = -4

By Vieta's formulas, the sum of the roots is equal to

-a

. The sum of the conjugate roots is

(2 + i\sqrt{b}) + (2 - i\sqrt{b}) = 4

, so

-a = 4

.

3

Calculate the constant term of the quadratic equation.

The constant term is

49

.

Substitute

a = -4

into the constant term

(a - 3)^2

to get

(-4 - 3)^2 = (-7)^2 = 49

.

4

Find the value of

b

using the product of the roots.

b = 45

By Vieta's formulas, the product of the roots is the constant term. The product is

(2 + i\sqrt{b})(2 - i\sqrt{b}) = 4 - i^2 b = 4 + b

(since

i^2 = -1

). Equating this to the constant term gives

4 + b = 49

, so

b = 45

.

5

Calculate the final value of

a + b

.

41

Substitute the values of

a

and

b

:

a + b = -4 + 45 = 41

.

Key Concept

Vieta's formulas and complex conjugate roots of quadratic equations with real coefficients.

Question 16Question →

If the quadratic equation $x^2 + kx + 25 = 0$ has exactly one real solution, and $k > 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

For any quadratic equation in standard form,

ax^2 + bx + c = 0

, the equation has exactly one real solution when the discriminant

b^2 - 4ac

is equal to

0

. Here, the equation is

x^2 + kx + 25 = 0

, so

a = 1

,

b = k

, and

c = 25

. Setting the discriminant to

0

gives

k^2 - 4(1)(25) = 0

, which simplifies to

k^2 - 100 = 0

. Solving for

k

yields

k = \pm 10

. Since we are given that

k > 0

, the only valid solution is

10

.

Step-by-Step Solution

1

Set the discriminant of the quadratic equation to zero.

b^2 - 4ac = 0

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

2

Identify coefficients

a

,

b

, and

c

from the equation

x^2 + kx + 25 = 0

and substitute them into the discriminant equation.

k^2 - 4(1)(25) = 0

For

x^2 + kx + 25 = 0

, the coefficients are

a=1

,

b=k

, and

c=25

.

3

Simplify the equation and solve for

k

.

k^2 = 100

, so

k = 10

or

k = -10

Simplifying

k^2 - 100 = 0

gives

k^2 = 100

.

4

Apply the constraint

k > 0

to find the final value.

k = 10

The problem specifies that

k

must be greater than zero.

Key Concept

Using the discriminant (

b^2 - 4ac

) of a quadratic equation to determine when there is exactly one real solution.

Estimated Time:45s

Question 17Question →

For a real constant $k$ , the quadratic equation $x^2 - 2(k - 2)x + (k^2 - 3k + 2) = 0$ has two real roots, $r_1$ and $r_2$ . What is the value of $k$ that minimizes the sum of the squares of these roots, $r_1^2 + r_2^2$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

that minimizes the sum of the squares of the roots is 2.

Applying Vieta's formulas and algebraic identities, the sum of the squares of the roots is expressed as

2k^2 - 10k + 12

. The discriminant condition for the roots to be real requires

k \leq 2

. Since the vertex of the upward-opening parabola

2k^2 - 10k + 12

is at

k = 2.5

, the function is strictly decreasing for all

k \leq 2

. Thus, the minimum value on the interval

k \leq 2

occurs at the boundary

k = 2

.

Step-by-Step Solution

1

Use Vieta's formulas to find the sum and product of the roots.

r_1 + r_2 = 2(k - 2)

and

r_1 r_2 = k^2 - 3k + 2

Vieta's formulas relate the roots of a quadratic equation

Ax^2 + Bx + C = 0

to its coefficients by

r_1 + r_2 = -B/A

and

r_1 r_2 = C/A

.

2

Express the sum of the squares of the roots,

r_1^2 + r_2^2

, in terms of

k

.

r_1^2 + r_2^2 = 2k^2 - 10k + 12

Since

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

, substituting the Vieta relations gives

[2(k-2)]^2 - 2(k^2 - 3k + 2) = 4(k^2 - 4k + 4) - 2k^2 + 6k - 4 = 2k^2 - 10k + 12

.

3

Determine the condition for the quadratic equation to have real roots using the discriminant.

\Delta = -4k + 8 \geq 0 \implies k \leq 2

For the roots

r_1

and

r_2

to be real, the discriminant

\Delta = B^2 - 4AC

must be greater than or equal to 0.

4

Minimize the quadratic function

f(k) = 2k^2 - 10k + 12

subject to the constraint

k \leq 2

.

k = 2

The parabola

f(k)

opens upwards with its vertex at

k = 2.5

. For

k \leq 2

, the function is strictly decreasing, meaning its minimum value on this interval occurs at the upper boundary,

k = 2

.

Key Concept

Quadratic Equations and the Quadratic Formula

Question 18Question →

For a certain quadratic equation $0.25x^2 - kx + 4.5 = 0$ , where $k$ is a constant, the difference between the two real solutions is exactly $3$ . What is the positive value of $k$ ?

Show answer & explanation

Answer: 2.25

Answer

The positive value of

k

is

2.25

.

By applying the root difference formula

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

with

a = 0.25

,

b = -k

, and

c = 4.5

, we get the equation

\frac{\sqrt{k^2 - 4.5}}{0.25} = 3

. This simplifies to

4\sqrt{k^2 - 4.5} = 3

. Dividing by

4

and squaring both sides gives

k^2 - 4.5 = 0.5625

, which means

k^2 = 5.0625

. Taking the positive square root yields

k = 2.25

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation and express the formula for the difference of the roots.

The coefficients are

a = 0.25

,

b = -k

, and

c = 4.5

. The difference between the roots is

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

.

This sets up the algebraic relationship between the root difference and the coefficients of the quadratic equation.

2

Substitute the coefficients and the given root difference of

3

into the difference formula.

\frac{\sqrt{(-k)^2 - 4(0.25)(4.5)}}{0.25} = 3

This creates an equation in terms of the variable

k

using the given problem constraints.

3

Simplify the equation and isolate the radical term.

4\sqrt{k^2 - 4.5} = 3 \implies \sqrt{k^2 - 4.5} = 0.75

Simplifying the fraction makes it easier to isolate the radical prior to squaring.

4

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

k^2

.

k^2 - 4.5 = 0.5625 \implies k^2 = 5.0625

Squaring is the inverse operation of the square root, allowing us to solve for

k^2

.

5

Calculate the positive square root of

5.0625

to find the value of

k

.

k = 2.25

Since the question asks for the positive value of

k

, we choose the positive square root.

Key Concept

Relating the difference of the roots of a quadratic equation to its coefficients using the discriminant and the quadratic formula.

Question 19Question →

For what value of the constant $k$ does the quadratic equation $0.5x^2 - 1.5x + k = 0$ have two complex roots of the form $a \pm 2i$ , where $a$ is a real number?

0.875

1.5625

2.125

3.125

6.25

Show answer & explanation

Answer: 3.125

Answer

3.125

Applying the quadratic formula to

0.5x^2 - 1.5x + k = 0

yields

x = 1.5 \pm \sqrt{2.25 - 2k}

. For the roots to be of the form

a \pm 2i

, the imaginary component must be

2i

, which requires the radicand

2.25 - 2k

to equal

(2i)^2 = -4

. Solving the equation

2.25 - 2k = -4

yields

k = 3.125

. This corresponds to the option containing

3.125

.

Step-by-Step Solution

1

Identify the coefficients

a

,

b

, and

c

of the given quadratic equation

0.5x^2 - 1.5x + k = 0

.

a = 0.5

,

b = -1.5

, and

c = k

.

These coefficients are required to substitute into the quadratic formula.

2

Apply the quadratic formula,

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

, to express the roots of the equation.

x = \frac{-(-1.5) \pm \sqrt{(-1.5)^2 - 4(0.5)k}}{2(0.5)} = \frac{1.5 \pm \sqrt{2.25 - 2k}}{1} = 1.5 \pm \sqrt{2.25 - 2k}

.

Substituting the coefficients simplifies the expression for the roots of the quadratic equation.

3

Set the radical term equal to the imaginary component

2i

to find the value of

k

that yields complex roots of the form

a \pm 2i

.

\sqrt{2.25 - 2k} = 2i \Rightarrow 2.25 - 2k = (2i)^2 \Rightarrow 2.25 - 2k = -4

.

For the roots to have an imaginary part of

2i

, the term under the square root must equal

(2i)^2

, which is

-4

.

4

Solve the linear equation

2.25 - 2k = -4

for

k

.

-2k = -4 - 2.25 \Rightarrow -2k = -6.25 \Rightarrow k = 3.125

.

Isolating

k

gives the constant value that satisfies the root condition.

Key Concept

Using the quadratic formula and discriminant to determine the value of a parameter for a specified type of complex roots.

Question 20Question →

A model rocket is launched vertically upward from a platform. Its height, $h$ , in meters, $t$ seconds after launch is modeled by the equation $h = -4.9t^2 + v_0 t + 2$ , where $v_0$ is the constant initial upward velocity in meters per second. If the rocket never reaches a height of $12$ meters, which of the following inequalities represents all possible values of $v_0$ ?

0 < v_0 < 7

0 < v_0 < 9.8

0 < v_0 < 14

0 < v_0 < 19.6

0 < v_0 < 28

Show answer & explanation

Answer: $0 < v_0 < 14$

Answer

0 < v_0 < 14

The correct answer is the range of values where the velocity is between 0 and 14. Setting the rocket's height equal to 12 meters gives the equation

-4.9t^2 + v_0 t + 2 = 12

. Subtracting 12 from both sides results in

-4.9t^2 + v_0 t - 10 = 0

. Since the rocket never reaches 12 meters, this quadratic equation has no real solutions, meaning its discriminant must be negative. Calculating the discriminant yields

v_0^2 - 4(-4.9)(-10) < 0

, which simplifies to

v_0^2 - 196 < 0

. Solving for a positive velocity gives

0 < v_0 < 14

.

Step-by-Step Solution

1

Set up the equation for the rocket reaching a height of 12 meters.

-4.9t^2 + v_0 t + 2 = 12

To find when the rocket reaches 12 meters, we set the height equation equal to 12.

2

Rearrange the equation into standard quadratic form.

-4.9t^2 + v_0 t - 10 = 0

Subtracting 12 from both sides of the equation sets it to 0, which is the standard form

at^2 + bt + c = 0

.

3

Apply the condition for no real solutions using the discriminant.

\Delta = v_0^2 - 4(-4.9)(-10) < 0

For the rocket to never reach 12 meters, the quadratic equation must have no real solutions. This requires the discriminant

\Delta = b^2 - 4ac

to be strictly negative.

4

Solve the inequality for the positive velocity

v_0

.

v_0^2 - 196 < 0 \implies v_0^2 < 196 \implies v_0 < 14

Multiplying the constants yields

4 \times 4.9 \times 10 = 196

. Taking the square root gives

v_0 < 14

. Since initial upward velocity must be positive,

0 < v_0 < 14

.

Key Concept

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

Quadratic Equations and the Quadratic Formula

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Answer

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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

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

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer