Intermediate Algebra Practice Questions — ACT — Page 13

Question 241Question →

The first term of an arithmetic sequence is $\frac{1}{2}$ , and the second term is $\frac{5}{6}$ . The first term of a geometric sequence is $2^2$ , and the common ratio is $2^3$ . Let $A$ be the third term of the arithmetic sequence, and let $G$ be the third term of the geometric sequence. If a third value, $V$ , is defined as $\frac{5}{6}$ less than twice $G$ , what is the value of $V + A$ ?

\frac{1541}{3}

512

\frac{1537}{3}

-510

\frac{24577}{3}

Show answer & explanation

Answer: $\frac{1537}{3}$

Answer

\frac{1537}{3}

The correct answer is

\frac{1537}{3}

because calculating the third term of the arithmetic sequence yields

A = \frac{7}{6}

and the third term of the geometric sequence yields

G = 256

. Translating the relationship for

V

yields

V = 2(256) - \frac{5}{6} = 512 - \frac{5}{6}

. Finding the sum of

V

and

A

yields

512 - \frac{5}{6} + \frac{7}{6} = 512\frac{1}{3} = \frac{1537}{3}

.

Step-by-Step Solution

1

Calculate the common difference

d

of the arithmetic sequence.

d = a_2 - a_1 = \frac{5}{6} - \frac{1}{2} = \frac{1}{3}

The common difference is the difference between any term and the preceding term in an arithmetic sequence.

2

Calculate the third term

A

of the arithmetic sequence.

A = a_1 + 2d = \frac{1}{2} + 2\left(\frac{1}{3}\right) = \frac{7}{6}

The

n

-th term of an arithmetic sequence is given by

a_n = a_1 + (n-1)d

.

3

Calculate the third term

G

of the geometric sequence.

G = g_1 \cdot r^2 = 2^2 \cdot (2^3)^2 = 2^2 \cdot 2^6 = 2^8 = 256

The

n

-th term of a geometric sequence is given by

g_n = g_1 \cdot r^{n-1}

.

4

Set up and solve for

V

using the algebraic relationship described.

V = 2G - \frac{5}{6} = 2(256) - \frac{5}{6} = 512 - \frac{5}{6}

The phrase '5/6 less than twice G' translates to

2G - \frac{5}{6}

.

5

Compute the sum of

V

and

A

.

V + A = \left(512 - \frac{5}{6}\right) + \frac{7}{6} = 512 + \frac{2}{6} = 512 + \frac{1}{3} = \frac{1537}{3}

Substitute the values of

V

and

A

and simplify the resulting fractional expression.

Key Concept

Arithmetic and Geometric Sequences and Series

Estimated Time:1m 30s

Question 242Question →

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 243Question →

A geometric sequence consists of positive terms and has a first term of $12$ and a common ratio of $r$ . An arithmetic sequence has a first term of $5$ and a common difference of $d$ . If the $3\text{rd}$ term of the geometric sequence is $3$ and the $4\text{th}$ term of the arithmetic sequence is $7$ , what is the value of $r + d$ ?

\frac{3}{5}

\frac{11}{8}

\frac{7}{6}

-\frac{15}{2}

\frac{35}{48}

Show answer & explanation

Answer: $\frac{7}{6}$

Answer

\frac{7}{6}

The correct value is

\frac{7}{6}

. This is found by first calculating the common ratio of the geometric sequence, where

12r^2 = 3 \Rightarrow r^2 = \frac{1}{4} \Rightarrow r = \frac{1}{2}

, and the common difference of the arithmetic sequence, where

5 + 3d = 7 \Rightarrow 3d = 2 \Rightarrow d = \frac{2}{3}

. Adding these two fractions with a common denominator yields

\frac{3}{6} + \frac{4}{6} = \frac{7}{6}

.

Step-by-Step Solution

1

Find the common ratio

r

of the geometric sequence.

r = \frac{1}{2}

The formula for the

n\text{th}

term of a geometric sequence is

g_n = g_1 \cdot r^{n-1}

. For the

3\text{rd}

term,

g_3 = 12r^2 = 3

, which simplifies to

r^2 = \frac{1}{4}

. Since the sequence has positive terms, we take the positive square root to get

r = \frac{1}{2}

.

2

Find the common difference

d

of the arithmetic sequence.

d = \frac{2}{3}

The formula for the

n\text{th}

term of an arithmetic sequence is

a_n = a_1 + (n-1)d

. For the

4\text{th}

term,

a_4 = 5 + 3d = 7

, which simplifies to

3d = 2

, or

d = \frac{2}{3}

.

3

Calculate the sum of

r

and

d

.

r + d = \frac{7}{6}

Adding the two values with a common denominator of

6

gives

\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}

.

Key Concept

Arithmetic and Geometric Sequences and Series

Estimated Time:1m 30s

Question 244Question →

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 245Question →

A sequence of numbers $t_1, t_2, t_3, \dots$ is defined by $t_1 = 3$ and $t_{n+1} = 3t_n - 2^n$ for all integers $n \geq 1$ . What is the value of the fourth term, $t_4$ ?

5

9

43

62

119

Show answer & explanation

Answer: 43

Answer

The value of the fourth term is 43.

To find

t_4

, we use the recursive formula

t_{n+1} = 3t_n - 2^n

. Substituting

n=1

gives

t_2 = 3(3) - 2 = 7

. Substituting

n=2

gives

t_3 = 3(7) - 4 = 17

. Substituting

n=3

gives

t_4 = 3(17) - 8 = 43

.

Step-by-Step Solution

1

Calculate the second term,

t_2

, by substituting

n = 1

and

t_1 = 3

into the formula

t_{n+1} = 3t_n - 2^n

.

t_2 = 3t_1 - 2^1 = 3(3) - 2 = 7

To progress to the fourth term, we must first find each preceding term in the sequence.

2

Calculate the third term,

t_3

, by substituting

n = 2

and

t_2 = 7

into the formula.

t_3 = 3t_2 - 2^2 = 3(7) - 4 = 17

Using the value of the second term allows us to find the third term.

3

Calculate the fourth term,

t_4

, by substituting

n = 3

and

t_3 = 17

into the formula.

t_4 = 3t_3 - 2^3 = 3(17) - 8 = 43

This completes the recursive process to find the target term.

Key Concept

Evaluating terms of a sequence defined by a recursive formula.

Estimated Time:1m 30s

Question 246Question →

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 247Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 = 25$ and a line is defined by the equation $3x + 4y = 15$ . The line intersects the circle at two points, $A$ and $B$ . What is the distance between point $A$ and point $B$ ?

Show answer & explanation

Answer: 8

Answer

The distance between the two intersection points is 8.

The correct answer is 8. We can find the intersection points by substituting

y = \frac{15 - 3x}{4}

into the circle's equation

x^2 + y^2 = 25

, yielding the quadratic equation

5x^2 - 18x - 35 = 0

. Solving this gives

x = 5

and

x = -1.4

, with corresponding

y

-coordinates

y = 0

and

y = 4.8

. The distance between

(5, 0)

and

(-1.4, 4.8)

is

\sqrt{(-1.4 - 5)^2 + (4.8 - 0)^2} = \sqrt{6.4^2 + 4.8^2} = \sqrt{64} = 8

. Alternatively, we can use geometry: the distance from the center of the circle

(0,0)

to the line

3x + 4y - 15 = 0

is

d = \frac{|3(0) + 4(0) - 15|}{\sqrt{3^2 + 4^2}} = 3

. Since the radius of the circle is

r = 5

, the right triangle formed by the radius, the perpendicular segment, and half the chord has a half-chord length of

\sqrt{5^2 - 3^2} = 4

. Thus, the total chord length is

2 \times 4 = 8

.

Step-by-Step Solution

1

Express the linear equation in terms of one variable

y = \frac{15 - 3x}{4}

This allows for substitution into the equation of the circle.

2

Substitute the expression into the circle's equation and simplify

x^2 + \left(\frac{15 - 3x}{4}\right)^2 = 25 \Rightarrow 16x^2 + (225 - 90x + 9x^2) = 400 \Rightarrow 25x^2 - 90x - 175 = 0 \Rightarrow 5x^2 - 18x - 35 = 0

To create a single quadratic equation in terms of

x

representing the intersection points.

3

Solve the quadratic equation for

x

(5x + 7)(x - 5) = 0 \Rightarrow x = 5

or

x = -1.4

To find the

x

-coordinates of the intersection points.

4

Calculate the corresponding

y

-coordinates

For

x = 5

,

y = 0

, giving point

A(5, 0)

. For

x = -1.4

,

y = 4.8

, giving point

B(-1.4, 4.8)

.

To determine the exact coordinates of both intersection points.

5

Apply the distance formula to find the length of the segment

AB

d = \sqrt{(-1.4 - 5)^2 + (4.8 - 0)^2} = \sqrt{(-6.4)^2 + 4.8^2} = \sqrt{40.96 + 23.04} = \sqrt{64} = 8

To compute the final distance between the two intersection points.

Key Concept

Solving systems of linear and quadratic equations to determine intersection points and calculating the distance between coordinates.

Question 248Question →

A geometric sequence of positive terms has a first term of $9$ and a third term of $4$ . What is the sum of the first $4$ terms of this sequence?

\frac{1}{3}

\frac{27}{4}

19

21

\frac{65}{3}

Show answer & explanation

Answer: $\frac{65}{3}$

Answer

The correct answer is

\frac{65}{3}

.

To find the sum of the first 4 terms of a geometric sequence with

a_1 = 9

and

a_3 = 4

, we first determine the common ratio

r

. Since

a_3 = a_1 \cdot r^2

, we have

4 = 9 \cdot r^2

, which gives

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

. Because the terms are positive,

r = \frac{2}{3}

. The first 4 terms are

9

,

6

,

4

, and

\frac{8}{3}

. Summing these terms gives

9 + 6 + 4 + \frac{8}{3} = 19 + \frac{8}{3} = \frac{65}{3}

.

Step-by-Step Solution

1

Find the common ratio

r

of the geometric sequence.

r = \frac{2}{3}

Since the sequence is geometric, the third term is related to the first term by

a_3 = a_1 \cdot r^2

. Substituting the given values yields

4 = 9 \cdot r^2

, which simplifies to

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

. Because all terms in the sequence are positive,

r

must be positive, so

r = \sqrt{\frac{4}{9}} = \frac{2}{3}

.

2

Calculate the first 4 terms of the sequence.

a_1 = 9

,

a_2 = 6

,

a_3 = 4

,

a_4 = \frac{8}{3}

Multiply each term by the common ratio

r = \frac{2}{3}

to find the subsequent term:

a_1 = 9

,

a_2 = 9 \cdot \frac{2}{3} = 6

,

a_3 = 6 \cdot \frac{2}{3} = 4

, and

a_4 = 4 \cdot \frac{2}{3} = \frac{8}{3}

.

3

Sum the first 4 terms of the sequence.

Sum =

\frac{65}{3}

Add the four terms:

9 + 6 + 4 + \frac{8}{3} = 19 + \frac{8}{3} = \frac{57}{3} + \frac{8}{3} = \frac{65}{3}

.

Key Concept

Calculating the sum of the first

n

terms of a geometric sequence given its first and third terms.

Question 249Question →

A geometric sequence has a first term of $\frac{1}{2}$ and a common ratio of $\frac{1}{4}$ . What is the sum of the first 3 terms of this sequence?

\frac{1}{14}

\frac{13}{16}

\frac{21}{32}

\frac{1}{32}

\frac{9}{128}

Show answer & explanation

Answer: $\frac{21}{32}$

Answer

The sum of the first 3 terms of this sequence is

\frac{21}{32}

.

The sum of the first three terms of a geometric sequence is calculated by finding each individual term and then adding them together. The first term is

\frac{1}{2}

. The second term is obtained by multiplying the first term by the common ratio:

\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}

. The third term is obtained by multiplying the second term by the common ratio:

\frac{1}{8} \times \frac{1}{4} = \frac{1}{32}

. To add these terms, we find a common denominator of 32:

\frac{16}{32} + \frac{4}{32} + \frac{1}{32} = \frac{21}{32}

.

Step-by-Step Solution

1

Identify the first three terms of the geometric sequence using the formula

a_n = a_1 \cdot r^{n-1}

.

The first term

a_1

is given as

\frac{1}{2}

. The second term is

a_2 = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}

. The third term is

a_3 = \frac{1}{8} \cdot \frac{1}{4} = \frac{1}{32}

.

Before calculating the sum, each individual term to be summed must be determined.

2

Find a common denominator to add the three fractional terms.

The least common multiple of the denominators 2, 8, and 32 is 32. Express the terms with this common denominator:

a_1 = \frac{16}{32}

,

a_2 = \frac{4}{32}

, and

a_3 = \frac{1}{32}

.

Adding fractions requires a common denominator.

3

Sum the adjusted fractions.

\frac{16}{32} + \frac{4}{32} + \frac{1}{32} = \frac{21}{32}

.

This yields the total sum of the first three terms.

Key Concept

Calculating the sum of a finite geometric series by finding and summing individual terms.

Question 250Question →

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 251Question →

A circle in the standard $(x, y)$ coordinate plane is described by the equation $x^2 + y^2 = 68$ . A line is described by the equation $y = x - 6$ . The line intersects the circle at two points. What is the distance between these two points of intersection?

8\sqrt{2}

2\sqrt{17}

10\sqrt{2}

20

6\sqrt{2}

Show answer & explanation

Answer: $10\sqrt{2}$

Answer

The distance between the two points of intersection is

10\sqrt{2}

.

To find the points of intersection, substitute

y = x - 6

into

x^2 + y^2 = 68

, which yields

x^2 + (x - 6)^2 = 68

. Expanding the binomial correctly results in

x^2 + x^2 - 12x + 36 = 68

. Combining like terms and writing the equation in standard form gives

2x^2 - 12x - 32 = 0

. Dividing the entire equation by 2 gives

x^2 - 6x - 16 = 0

. Factoring this quadratic equation yields

(x - 8)(x + 2) = 0

, giving solutions

x = 8

and

x = -2

. Substituting these values back into the linear equation gives the points

(8, 2)

and

(-2, -8)

. Finally, using the distance formula, the distance between the two points is

\sqrt{(8 - (-2))^2 + (2 - (-8))^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}

.

Step-by-Step Solution

1

Substitute the linear equation into the circle equation.

x^2 + (x - 6)^2 = 68

To eliminate

y

and solve for the

x

-coordinates of the intersection points.

2

Expand the binomial and collect like terms.

2x^2 - 12x - 32 = 0

Expanding

(x - 6)^2

yields

x^2 - 12x + 36

, and combining it with

x^2

and subtracting

68

from both sides puts the equation in quadratic form.

3

Divide the quadratic equation by 2 and solve by factoring.

(x - 8)(x + 2) = 0

, so

x = 8

or

x = -2

Simplifying the equation to

x^2 - 6x - 16 = 0

makes it easy to find the roots of the quadratic equation.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into the linear equation.

For

x = 8

,

y = 2

, yielding point

(8, 2)

. For

x = -2

,

y = -8

, yielding point

(-2, -8)

.

To determine the full coordinates of the two intersection points.

5

Apply the distance formula to find the distance between

(8, 2)

and

(-2, -8)

.

d = \sqrt{(8 - (-2))^2 + (2 - (-8))^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}

To calculate the straight-line distance between the two coordinate points.

Key Concept

Solving systems of linear and non-linear equations by substitution and finding the distance between intersection points.

Estimated Time:1m 30s

Question 252Question →

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 253Question →

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 254Question →

In the standard $(x, y)$ coordinate system, a line given by the equation $y = 2x + 4$ intersects a parabola given by the equation $y = x^2 - 2x - 1$ at exactly two points. What is the sum of the $y$ -coordinates of these two points of intersection?

Show answer & explanation

Answer: 16

Answer

16

The system of equations is solved by setting the equations equal to each other, yielding the quadratic equation

x^2 - 4x - 5 = 0

. Solving for

x

gives

x = 5

and

x = -1

. Substituting these values into the linear equation gives the y-coordinates

14

and

2

. The sum of these y-coordinates is

14 + 2 = 16

.

Step-by-Step Solution

1

Equate the linear and quadratic equations to find the x-values of the intersection points.

x^2 - 2x - 1 = 2x + 4

At the points of intersection, the y-values of both equations must be equal.

2

Rearrange the equation to standard quadratic form.

x^2 - 4x - 5 = 0

Grouping all terms on one side allows the quadratic equation to be solved.

3

Factor the quadratic equation to find the x-coordinates.

(x - 5)(x + 1) = 0

, so

x = 5

or

x = -1

The numbers that multiply to

-5

and add up to

-4

are

-5

and

1

.

4

Substitute the x-coordinates into the linear equation to find the y-coordinates.

For

x = 5

:

y = 2(5) + 4 = 14

. For

x = -1

:

y = 2(-1) + 4 = 2

.

Evaluating the linear equation is simpler than evaluating the quadratic equation.

5

Find the sum of the y-coordinates.

14 + 2 = 16

The question asks for the sum of the y-coordinates of the two points of intersection.

Key Concept

Solving a system of linear and quadratic equations by substitution.

Alternative Method

Instead of solving for the individual intersection points, Vieta's formulas can be applied. The x-coordinates satisfy

x^2 - 4x - 5 = 0

, so their sum is

x_1 + x_2 = 4

. Since the points lie on the line

y = 2x + 4

, the sum of the y-coordinates is

y_1 + y_2 = (2x_1 + 4) + (2x_2 + 4) = 2(x_1 + x_2) + 8 = 2(4) + 8 = 16

.

Estimated Time:1m 30s

Question 255Question →

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 256Question →

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 257Question →

The path of a particle in the standard $(x, y)$ coordinate plane is described by the linear equation $3x - y = 2$ , and the path of another particle is described by the quadratic equation $y = x^2 - x - 7$ . If the two paths intersect at two locations, what is the sum of the $y$ -coordinates of these intersection points?

Show answer & explanation

Answer: 8

Answer

The sum of the

y

-coordinates of the intersection points is 8.

The correct answer is 8. Solving the system by setting

x^2 - x - 7 = 3x - 2

results in the quadratic equation

x^2 - 4x - 5 = 0

. Factoring gives

(x-5)(x+1) = 0

, which yields intersection

x

-coordinates of

x = 5

and

x = -1

. Substituting these back into the linear equation gives

y

-coordinates of

13

and

-5

. Adding these values together yields

13 + (-5) = 8

.

Step-by-Step Solution

1

Express the linear equation in terms of

y

.

y = 3x - 2

This allows for direct substitution into the quadratic equation.

2

Equate the linear and quadratic expressions to solve for the

x

-coordinates of the intersection points.

x^2 - x - 7 = 3x - 2

Intersection points share the same coordinates for both equations.

3

Set the quadratic equation to zero.

x^2 - 4x - 5 = 0

This puts the equation in standard form so it can be solved by factoring.

4

Factor the quadratic equation.

(x - 5)(x + 1) = 0

, so

x = 5

or

x = -1

Factoring determines the

x

-coordinates of the intersection points.

5

Substitute the

x

-values into the linear equation to determine the

y

-coordinates.

For

x = 5

,

y = 13

. For

x = -1

,

y = -5

.

Finding the

y

-coordinates is necessary to compute their sum.

6

Add the

y

-coordinates together.

13 + (-5) = 8

The question asks for the sum of the

y

-coordinates of the intersection points.

Key Concept

Solving systems of linear and quadratic equations by substitution and factoring

Alternative Method

We can use Vieta's formulas to find the sum of the

y

-coordinates without calculating each individual coordinate. The sum of the

y

-coordinates is

y_1 + y_2 = (3x_1 - 2) + (3x_2 - 2) = 3(x_1 + x_2) - 4

. Since

x_1

and

x_2

are the roots of

x^2 - 4x - 5 = 0

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

x_1 + x_2 = -\frac{-4}{1} = 4

. Substituting this value into our sum expression yields

3(4) - 4 = 12 - 4 = 8

.

Estimated Time:1m 30s

Question 258Question →

A parabolic arch is modeled by the equation $y = (x - 3)^2 - 5$ in the standard $(x, y)$ coordinate plane. A straight pathway is modeled by a line where the $y$ -coordinate of any point is 2 less than its $x$ -coordinate. If the pathway intersects the arch at points $A$ and $B$ , what is the area, in square units, of the triangle with vertices at $A$ , $B$ , and the origin $(0, 0)$ ?

3

5

7

8

10

Show answer & explanation

Answer: 5

Answer

The area of the triangle with vertices at the intersection points and the origin is 5 square units.

To find the area of the triangle, we first solve the system of equations. Substituting the pathway's equation y = x - 2 into the parabola's equation y = (x - 3)^2 - 5 yields x^2 - 7x + 6 = 0, which factors to (x - 6)(x - 1) = 0. This gives x = 6 and x = 1. The corresponding y-coordinates are y = 4 and y = -1, representing the intersection points (6, 4) and (1, -1). The area of the triangle with these vertices and the origin (0, 0) is calculated as 0.5 * |6(-1) - 4(1)| = 5.

Step-by-Step Solution

1

Set up the system of equations by substituting the linear equation into the quadratic equation.

(x - 3)^2 - 5 = x - 2

The pathway is described as having a y-coordinate that is 2 less than the x-coordinate, which translates to the linear equation y = x - 2. Substituting this into the parabola's equation allows us to find the intersection points.

2

Expand the quadratic term and simplify the equation into standard quadratic form.

x^2 - 7x + 6 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting

x

and adding

2

to both sides results in standard quadratic form.

3

Factor the quadratic equation to solve for the x-coordinates.

(x - 6)(x - 1) = 0

, so

x = 6

or

x = 1

Factoring allows us to find the x-values that satisfy the intersection condition.

4

Substitute the x-values back into the linear equation to find the corresponding y-coordinates.

For

x = 6

,

y = 4

giving point

(6, 4)

. For

x = 1

,

y = -1

giving point

(1, -1)

.

The intersection points must satisfy both equations in the system.

5

Calculate the area of the triangle with vertices

(0, 0)

,

(6, 4)

, and

(1, -1)

using the coordinate area formula.

Area =

\frac{1}{2} |6(-1) - 4(1)| = \frac{1}{2} |-10| = 5

square units.

The area of a triangle with one vertex at the origin and others at

(x_1, y_1)

and

(x_2, y_2)

is

\frac{1}{2} |x_1 y_2 - y_1 x_2|

.

Key Concept

Solving systems of linear and quadratic equations and finding the area of a triangle in the coordinate plane.

Question 259Question →

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 260Question →

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

Intermediate Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Alternative Method

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

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept