All practice questions

For what value of $k$ does the equation $\frac{1}{5}(kx - 3) - \frac{1}{3}(2x - 5) = 2$ have a solution of $x = 7$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Substituting

x = 7

reduces the equation to

\frac{1}{5}(7k - 3) - 3 = 2

. Adding 3 to both sides yields

\frac{1}{5}(7k - 3) = 5

. Multiplying by 5 gives

7k - 3 = 25

. Adding 3 gives

7k = 28

, which results in

k = 4

.

Step-by-Step Solution

1

Substitute

x = 7

into the equation.

\frac{1}{5}(7k - 3) - \frac{1}{3}(2(7) - 5) = 2

Since

x = 7

is given as a solution, substituting it into the equation must make the equality true.

2

Evaluate and simplify the expression in the second term.

\frac{1}{3}(14 - 5) = \frac{1}{3}(9) = 3

Follow the order of operations by simplifying the expression inside the parentheses first.

3

Isolate the fractional term containing the variable

k

.

\frac{1}{5}(7k - 3) - 3 = 2 \implies \frac{1}{5}(7k - 3) = 5

Add 3 to both sides of the equation to eliminate the subtraction of 3.

4

Clear the fraction and solve the remaining linear equation for

k

.

7k - 3 = 25 \implies 7k = 28 \implies k = 4

Multiply both sides by 5 to eliminate the denominator, add 3 to isolate the term with

k

, and divide by 7.

Key Concept

Solving multi-step linear equations containing parameters and fractions.

Estimated Time:1m 30s

Question 122Question →

Let matrix $A = \begin{pmatrix} 2 & 5 \\ -1 & 3 \end{pmatrix}$ and matrix $B = \begin{pmatrix} 4 & -2 \\ 3 & 1 \end{pmatrix}$ . If matrix $C = 2A + 3B$ , what is the value of the entry in the second row and first column of matrix $C$ ?

Show answer & explanation

Answer: 7

Answer

The entry in the second row and first column of matrix

C

is 7.

The entry in the second row, first column of matrix

C

, denoted as

c_{2,1}

, is calculated by multiplying the corresponding entry in matrix

A

(

a_{2,1} = -1

) by 2 and the corresponding entry in matrix

B

(

b_{2,1} = 3

) by 3, and then adding the results:

2(-1) + 3(3) = -2 + 9 = 7

.

Step-by-Step Solution

1

Locate the element in the second row, first column of both matrices

A

and

B

.

The element in the second row, first column of

A

(designated

a_{2,1}

) is

-1

, and the element in the second row, first column of

B

(designated

b_{2,1}

) is

3

.

To find a specific entry in the resulting matrix

C = 2A + 3B

, we perform the scalar multiplication and addition operations on the corresponding entries in the same position of matrices

A

and

B

.

2

Multiply the corresponding elements by their respective scalar factors.

For matrix

A

,

2 \times (-1) = -2

. For matrix

B

,

3 \times 3 = 9

.

Scalar multiplication requires multiplying each individual entry of a matrix by the given scalar factor.

3

Add the two resulting values to find the entry for matrix

C

.

-2 + 9 = 7

Matrix addition is performed by adding the corresponding elements of the matrices.

Key Concept

Matrix scalar multiplication and addition of corresponding elements

Estimated Time:45s

Question 123Question →

The algebraic expression $\frac{(x^2)^a}{x^{-4}}$ is equivalent to $x^{10}$ for all non-zero real numbers $x$ . What is the value of $a$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

The correct value for

a

is

3

. Applying the power of a power rule to

(x^2)^a

yields

x^{2a}

. Then, applying the quotient rule to

\frac{x^{2a}}{x^{-4}}

yields

x^{2a - (-4)} = x^{2a+4}

. Equating the exponents gives

2a + 4 = 10

, which solves to

a = 3

.

Step-by-Step Solution

1

Apply the power of a power property to the numerator.

(x^2)^a = x^{2a}

When raising a power to another power, multiply the exponents:

(x^m)^n = x^{mn}

.

2

Apply the quotient property of exponents to simplify the fraction.

\frac{x^{2a}}{x^{-4}} = x^{2a - (-4)} = x^{2a + 4}

When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator:

\frac{x^m}{x^n} = x^{m-n}

.

3

Set the resulting exponent equal to the exponent of the equivalent expression and solve for

a

.

2a + 4 = 10 \implies 2a = 6 \implies a = 3

Since the bases are identical and the expressions are equivalent, their exponents must be equal.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 124Question →

A line with a positive slope passes through the point $(0, -4)$ and is tangent to the circle $x^2 + y^2 = 4$ . If this same line is also tangent to the parabola $y = x^2 + k$ , what is the value of the constant $k$ ?

Show answer & explanation

Answer: -3.25

Answer

The constant

k

must be

-3.25

.

By writing the equation of the line passing through

(0, -4)

as

y = mx - 4

and applying the condition that it is tangent to the circle

x^2 + y^2 = 4

, we find the positive slope is

m = \sqrt{3}

. Substituting this tangent line

y = \sqrt{3}x - 4

into the parabola equation

y = x^2 + k

yields the quadratic equation

x^2 - \sqrt{3}x + (k + 4) = 0

. For the line to be tangent to the parabola, the discriminant of this equation must be zero, which gives

3 - 4(k + 4) = 0

, leading to the final value

k = -3.25

.

Step-by-Step Solution

1

Set up the equation of the line passing through

(0, -4)

and substitute it into the equation of the circle.

x^2 + (mx - 4)^2 = 4 \implies (1 + m^2)x^2 - 8mx + 12 = 0

This establishes the quadratic equation representing the intersection points of the line and the circle.

2

Solve for the slope

m

by setting the discriminant of the intersection equation to zero.

(-8m)^2 - 4(1 + m^2)(12) = 0 \implies 64m^2 - 48 - 48m^2 = 0 \implies 16m^2 = 48 \implies m = \sqrt{3}

Since the line is tangent to the circle, the intersection equation must have exactly one real solution, meaning its discriminant is zero. The positive root is selected because the slope is specified as positive.

3

Equate the line

y = \sqrt{3}x - 4

and the parabola

y = x^2 + k

to analyze their intersection.

x^2 - \sqrt{3}x + (k + 4) = 0

This sets up the quadratic equation for the intersection points between the tangent line and the parabola.

4

Solve for

k

by setting the discriminant of this second quadratic equation to zero.

(-\sqrt{3})^2 - 4(1)(k + 4) = 0 \implies 3 - 4k - 16 = 0 \implies -13 - 4k = 0 \implies k = -3.25

For the line to be tangent to the parabola, the system must have exactly one solution, meaning the discriminant of their intersection equation must be zero.

Key Concept

Systems of Linear and Non-Linear Equations

Question 125Question →

Let the functions $f$ and $g$ be defined by $f(x) = \sqrt{3x + 4}$ and $g(x) = x^2 - 2$ . What is the positive value of $x$ for which the composite function value $f(g(x)) = 5$ ?

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. We set

f(g(x)) = 5

, which translates to

\sqrt{3g(x) + 4} = 5

. Squaring both sides yields

3g(x) + 4 = 25

. Solving for

g(x)

gives

3g(x) = 21

, which simplifies to

g(x) = 7

. Substituting

g(x) = x^2 - 2

into this equation gives

x^2 - 2 = 7

, which simplifies to

x^2 = 9

. Taking the square root of both sides gives

x = 3

or

x = -3

. Since the question specifies the positive value of

x

, the correct answer is 3.

Step-by-Step Solution

1

Substitute the inner function

g(x)

into the outer function

f(x)

to express

f(g(x)) = 5

\sqrt{3g(x) + 4} = 5

Setting up the initial composite equation to isolate the inner function

g(x)

2

Square both sides of the equation and isolate

g(x)

3g(x) + 4 = 25 \implies 3g(x) = 21 \implies g(x) = 7

Eliminating the radical allows us to determine the required output value of

g(x)

3

Substitute the formula for

g(x)

into the equation and solve for

x

x^2 - 2 = 7 \implies x^2 = 9 \implies x = 3

or

x = -3

Finding the input values of

x

that satisfy the condition for

g(x)

4

Select the positive value from the solution set

x = 3

The question specifically requests the positive value of

x

Key Concept

Evaluating and solving composite function equations by working from the outside function inward

Alternative Method

Instead of solving from the outside in, we can first construct the symbolic expression for

f(g(x))

by substituting

g(x)

directly:

f(g(x)) = \sqrt{3(x^2 - 2) + 4} = \sqrt{3x^2 - 6 + 4} = \sqrt{3x^2 - 2}

. Setting this equal to 5 yields

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

. Squaring both sides gives

3x^2 - 2 = 25 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = \pm 3

. The positive solution is 3.

Estimated Time:1m 30s

Question 126Question →

An online store sells digital songs for $\$1.20$ each. A customer uses a discount code to get $\$3.00$ off the total purchase. If the customer's total cost after using the discount code is $\$15.00$ , how many digital songs did the customer purchase?

Show answer & explanation

Answer: 15

Answer

The customer purchased 15 songs.

The word problem translates directly to the linear equation

1.20s - 3.00 = 15.00

, where

s

is the number of songs. Adding

3.00

to both sides of the equation yields

1.20s = 18.00

. Dividing both sides by

1.20

yields

s = 15

.

Step-by-Step Solution

1

Define the variable and set up the equation.

Let

s

be the number of digital songs purchased. The cost of

s

songs is

1.20s

. Subtracting the discount of

\$3.00

gives the equation:

1.20s - 3.00 = 15.00

.

We must represent the cost of the songs and the discount algebraically to equal the final payment.

2

Isolate the variable term.

1.20s = 18.00

Adding

3.00

to both sides of the equation simplifies the equation by canceling the subtraction of

3.00

.

3

Solve for the variable.

s = 15

Dividing both sides of the equation by

1.20

isolates

s

to find the total number of songs.

Key Concept

Translating real-world scenarios with linear relationships into algebraic equations and solving them.

Question 127Question →

For all positive real numbers

x

and

y

, the expression

\frac{(3x^2 y^{-3})^3 \cdot (2x^{-1} y^2)^2}{(6x^3 y^{-2})^2}

can be simplified to the form

A x^a y^b

, where

A

,

a

, and

b

are integers. What is the value of the sum

A + a + b

?

Show answer & explanation

Answer: 0

Answer

The value of the sum

A + a + b

is 0.

By applying the rules of exponents systematically, the expression simplifies to

3 x^{-2} y^{-1}

. Comparing this to

A x^a y^b

yields

A = 3

,

a = -2

, and

b = -1

. The sum is

3 + (-2) + (-1) = 0

.

Step-by-Step Solution

1

Simplify the first term in the numerator

27x^6y^{-9}

Apply the power of a product rule and power of a power rule to

(3x^2 y^{-3})^3

.

2

Simplify the second term in the numerator

4x^{-2}y^4

Apply the power of a product rule and power of a power rule to

(2x^{-1} y^2)^2

.

3

Multiply the simplified terms in the numerator together

108x^4y^{-5}

Multiply coefficients and add the exponents of like bases.

4

Simplify the denominator

36x^6y^{-4}

Apply the power of a product rule and power of a power rule to

(6x^3 y^{-2})^2

.

5

Divide the numerator by the denominator

3x^{-2}y^{-1}

Divide the coefficients and subtract the denominator exponents from the numerator exponents for like bases.

6

Sum the constants

A

,

a

, and

b

0

Identify

A = 3

,

a = -2

,

b = -1

from the expression

3x^{-2}y^{-1}

, and calculate

3 + (-2) + (-1) = 0

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 128Question →

If $2.5(x - 4) = 15$ , what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The value of

x

is

10

.

To solve the equation

2.5(x - 4) = 15

, divide both sides by

2.5

to obtain

x - 4 = 6

. Then, add

4

to both sides to get the final solution of

x = 10

. Alternatively, distribute

2.5

to get

2.5x - 10 = 15

, add

10

to both sides to get

2.5x = 25

, and divide by

2.5

to get

x = 10

.

Step-by-Step Solution

1

Divide both sides of the equation by

2.5

.

x - 4 = 6

To isolate the parenthetical term on the left side of the equation.

2

Add

4

to both sides of the equation.

x = 10

To isolate the variable

x

.

Key Concept

Solving multi-step linear equations using inverse operations.

Alternative Method

Distribute

2.5

to get

2.5x - 10 = 15

. Add

10

to both sides of the equation to get

2.5x = 25

. Divide both sides by

2.5

to find

x = 10

.

Estimated Time:45s

Question 129Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 = 10$ . The line $y = 3x$ intersects the circle at a point $(x, y)$ in the first quadrant. What is the value of $x + y$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Substituting

y = 3x

into the equation

x^2 + y^2 = 10

yields

x^2 + (3x)^2 = 10

. Simplifying this expression gives

x^2 + 9x^2 = 10

, which combines to

10x^2 = 10

. Dividing both sides by 10 results in

x^2 = 1

. Since the point of intersection lies in the first quadrant, we take the positive square root, so

x = 1

. Substituting

x = 1

back into

y = 3x

gives

y = 3(1) = 3

. The sum of the coordinates is

x + y = 1 + 3 = 4

.

Step-by-Step Solution

1

Substitute the linear expression for

y

into the equation of the circle.

x^2 + (3x)^2 = 10

Substituting one equation into another allows us to solve for a single variable.

2

Simplify the equation and solve for

x

.

10x^2 = 10 \Rightarrow x^2 = 1 \Rightarrow x = 1

Combining like terms gives

10x^2 = 10

, which simplifies to

x^2 = 1

. Since the intersection point is in the first quadrant,

x

must be positive.

3

Substitute the value of

x

back into the linear equation to find

y

.

y = 3(1) = 3

Using the value of

x

in the linear equation determines the corresponding

y

-coordinate.

4

Calculate the sum of the coordinates

x + y

.

1 + 3 = 4

The question asks for the value of the sum of the coordinates of the intersection point.

Key Concept

Solving a system of linear and non-linear (circular) equations using substitution.

Question 130Question →

If the polynomial $2x^2 + kx - 12$ can be factored as the product of two binomials with integer coefficients, what is the greatest possible integer value of $k$ ?

Show answer & explanation

Answer: 23

Answer

The greatest possible integer value of

k

is 23.

The greatest possible value of

k

is 23. This is found by setting the factored form of the expression as

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

. We find that

ab = -12

and

k = 2b + a

. The possible integer pairs for

(a, b)

that multiply to

-12

are tested to find the maximum sum of

2b + a

. The maximum is achieved when

b = 12

and

a = -1

, yielding

2(12) - 1 = 23

.

Step-by-Step Solution

1

Set up the algebraic representation for the binomial factors with integer coefficients.

The polynomial must factor into the form

(2x + a)(x + b)

where

a

and

b

are integers.

Since the leading coefficient is 2 (a prime number), the coefficients of the linear terms in the binomial factors must be 2 and 1 to obtain a product of

2x^2

.

2

Expand the product of the binomials to relate the parameters

a

and

b

to the original expression.

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

, which implies

ab = -12

and

k = 2b + a

.

By equating the coefficients of corresponding terms in the expanded expression and the original polynomial, we establish relationships for the constant term and the linear coefficient.

3

Analyze the factors of

-12

to find the integer values of

a

and

b

that maximize the linear term coefficient.

Choosing

b = 12

and

a = -1

gives

ab = -12

and results in

k = 2(12) + (-1) = 23

.

To maximize

2b + a

, we choose the largest possible positive factor of

-12

for the term multiplied by 2, which is 12, paired with the corresponding negative factor

-1

for

a

.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

with

a > 1

Question 131Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{(1 + 2i)^3}{2 - i}$ . What is the absolute value of $z$ ?

Show answer & explanation

Answer: 5

Answer

5

The complex number

z

simplifies to

-4 - 3i

. The absolute value of a complex number

a + bi

is defined as

\sqrt{a^2 + b^2}

, which gives

\sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5

. Alternatively, using properties of the modulus,

|z| = \frac{|1 + 2i|^3}{|2 - i|} = \frac{(\sqrt{1^2 + 2^2})^3}{\sqrt{2^2 + (-1)^2}} = \frac{(\sqrt{5})^3}{\sqrt{5}} = 5

.

Step-by-Step Solution

1

Expand the numerator

(1 + 2i)^3

-11 - 2i

To begin simplifying the complex fraction by expanding the cubed binomial in the numerator.

2

Divide the result by the denominator

2 - i

by multiplying both numerator and denominator by the complex conjugate

2 + i

-4 - 3i

To express the quotient in standard complex form

a + bi

.

3

Find the absolute value of the simplified complex number

z = -4 - 3i

5

To calculate the final magnitude of the complex number using

|a + bi| = \sqrt{a^2 + b^2}

.

Key Concept

Complex number operations, complex division, and absolute value (modulus) of complex numbers

Question 132Question →

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 133Question →

A company offers two monthly data storage plans. Plan A costs $32.00 plus$ 0.08 per gigabyte of data stored. Plan B costs $15.00 plus$ 0.14 per gigabyte for the first 150 gigabytes of data stored, and $0.12 per gigabyte for all data stored beyond the first 150 gigabytes. For how many gigabytes of data stored in a single month would the monthly cost under both plans be exactly the same?

Show answer & explanation

Answer: 350

Answer

The monthly costs under both plans are exactly the same when the data stored is 350 gigabytes.

Equating the two cost functions yields

32 + 0.08g = 15 + 0.14(150) + 0.12(g - 150)

, which simplifies to

32 + 0.08g = 18 + 0.12g

. Solving for

g

gives

0.04g = 14

, which results in

g = 350

.

Step-by-Step Solution

1

Determine the cost function for Plan A.

C_A(g) = 32 + 0.08g

Plan A charges a flat fee of

32.00 and

0.08 per gigabyte.

2

Determine the cost function for Plan B when the usage exceeds 150 gigabytes.

C_B(g) = 15 + 0.14(150) + 0.12(g - 150) = 18 + 0.12g

Plan B charges

15.00 flat,

0.14 per gigabyte for the first 150 gigabytes, and $0.12 per gigabyte for any additional usage.

3

Equate the two cost functions and solve for

g

.

32 + 0.08g = 18 + 0.12g \implies 14 = 0.04g \implies g = 350

To find the usage where both plans cost the same, set their cost functions equal and isolate the variable

g

.

Key Concept

Solving linear equations in real-life contexts involving piecewise rates

Question 134Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq 3$ , what is the real solution to the equation $\frac{x}{x-3} + \frac{2}{x} = \frac{3}{x-3}$ ?

Show answer & explanation

Answer: -2

Answer

The correct answer is -2.

Subtracting

\frac{x}{x-3}

from both sides yields

\frac{2}{x} = \frac{3-x}{x-3}

. Since

3-x = -(x-3)

, the right side simplifies to

-1

for all

x \neq 3

. The equation becomes

\frac{2}{x} = -1

, which gives

x = -2

. Since

-2

does not violate the domain constraints, it is the correct solution.

Step-by-Step Solution

1

Subtract

\frac{x}{x-3}

from both sides of the equation.

\frac{2}{x} = \frac{3-x}{x-3}

To group terms with common denominators on the same side.

2

Simplify the fraction on the right side of the equation.

\frac{2}{x} = -1

The numerator

3-x

is the negative of the denominator

x-3

, so their quotient is

-1

for all

x \neq 3

.

3

Solve the simplified equation

\frac{2}{x} = -1

for

x

.

x = -2

Multiplying both sides by

x

yields

2 = -x

, and dividing by

-1

gives

x = -2

.

4

Check the solution against the domain restrictions

x \neq 0

and

x \neq 3

.

The solution

x = -2

is valid.

Since

-2

is neither

0

nor

3

, it does not cause any denominator in the original equation to equal zero.

Key Concept

Solving rational equations by isolating terms with common denominators and checking for extraneous solutions.

Question 135Question →

A large rectangle has a length of $4x + 3$ inches and a width of $3x - 2$ inches. A smaller rectangular region is cut out from the center. The cut-out region has a length of $2x - 1$ inches and a width of $x - 5$ inches. The area of the remaining region can be written as the polynomial $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12, which is the coefficient of the linear term in the simplified remaining area polynomial.

The remaining area is found by subtracting the area of the smaller rectangle from the area of the larger rectangle. The area of the larger rectangle is

(4x + 3)(3x - 2) = 12x^2 + x - 6

. The area of the smaller rectangle is

(2x - 1)(x - 5) = 2x^2 - 11x + 5

. Subtracting the two yields

(12x^2 + x - 6) - (2x^2 - 11x + 5) = 10x^2 + 12x - 11

. Matching this to

Ax^2 + Bx + C

, the coefficient

B

of the linear term is

12

.

Step-by-Step Solution

1

Find the area of the outer rectangle.

Area =

12x^2 + x - 6

Multiply the length

(4x + 3)

and the width

(3x - 2)

using the distributive property.

2

Find the area of the inner rectangle.

Area =

2x^2 - 11x + 5

Multiply the length

(2x - 1)

and the width

(x - 5)

using the distributive property.

3

Subtract the inner area from the outer area to find the remaining area.

Remaining Area =

10x^2 + 12x - 11

Subtract the polynomial

(2x^2 - 11x + 5)

from

(12x^2 + x - 6)

by distributing the negative sign and combining like terms.

4

Identify the coefficient

B

of the

x

term.

B = 12

Match the simplified polynomial

10x^2 + 12x - 11

with the form

Ax^2 + Bx + C

.

Key Concept

Operations on Polynomials (multiplication and subtraction of polynomials)

Question 136Question →

A landscaping company charges a flat equipment fee plus a fixed hourly rate for lawn maintenance. For a job that took $4$ hours, the company charged a total of $\$190$ . For a different job that took $7$ hours, the company charged a total of $\$295$ . The company also offers a package that includes a $15\%$ discount on the hourly rate, but the flat equipment fee remains the same. Under this discounted package, what is the total charge, in dollars, for a job that takes $8$ hours?

Show answer & explanation

Answer: 288

Answer

The total charge for an

8

-hour job under the discounted package is

288

dollars.

By translating the problem into the equations

F + 4H = 190

and

F + 7H = 295

, we find that the hourly rate

H

is

35

dollars and the flat fee

F

is

50

dollars. Applying a

15\%

discount to the hourly rate yields a new rate of

29.75

dollars per hour. The total cost for

8

hours is then

50 + 8(29.75) = 288

dollars.

Step-by-Step Solution

1

Define variables for the flat fee and hourly rate, and translate the given information into a system of linear equations.

F + 4H = 190

and

F + 7H = 295

, where

F

is the flat fee and

H

is the hourly rate.

To represent the cost structure algebraically.

2

Solve the system of equations by elimination or substitution.

H = 35

and

F = 50

.

To determine the individual cost components (flat fee and hourly rate).

3

Apply the

15\%

discount to the hourly rate.

H_{\text{discounted}} = 35 \times (1 - 0.15) = 29.75

.

To find the new hourly rate under the discounted package.

4

Calculate the total cost for

8

hours of work with the flat fee and the discounted hourly rate.

\text{Total Cost} = 50 + 8(29.75) = 288

.

To answer the question asking for the total charge of an

8

-hour job under the discounted package.

Key Concept

Translating verbal statements into a system of linear equations and solving them to evaluate a modified expression.

Question 137Question →

When the expression $(5x^2 - 8x + 2) - (2x^2 - 4x - 9)$ is written in the standard form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of $b$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

, which is the coefficient of the

x

term after simplifying the expression.

To subtract polynomials, we distribute the negative sign to all terms of the polynomial being subtracted and then combine like terms. Simplifying

(5x^2 - 8x + 2) - (2x^2 - 4x - 9)

gives

5x^2 - 8x + 2 - 2x^2 + 4x + 9 = 3x^2 - 4x + 11

. Comparing this to

ax^2 + bx + c

shows that

b

, the coefficient of the

x

term, is

-4

.

Step-by-Step Solution

1

Distribute the subtraction sign to all terms inside the second set of parentheses.

-2x^2 + 4x + 9

Subtracting a polynomial is equivalent to adding its opposite, which means changing the sign of every term in that polynomial.

2

Group and combine like terms from both polynomials.

3x^2 - 4x + 11

Like terms (terms with the same variable raised to the same power) can be combined by adding or subtracting their coefficients:

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

,

(-8x + 4x) = -4x

, and

(2 + 9) = 11

.

3

Compare the simplified expression to the standard form

ax^2 + bx + c

to identify the value of

b

.

b = -4

In the standard form

ax^2 + bx + c

, the coefficient of the

x

term is represented by

b

. In the simplified expression

3x^2 - 4x + 11

, the coefficient of

x

is

-4

.

Key Concept

Polynomial Subtraction and Combining Like Terms

Question 138Question →

For the functions $f(x) = \frac{18}{x - 2}$ and $g(x) = x^2 - 1$ , what is the value of $f(g(3))$ ?

Show answer & explanation

Answer: 3

Answer

The value of the composite function

f(g(3))

is 3.

To evaluate the composite function

f(g(3))

, we start by evaluating the inner function

g(3) = 3^2 - 1 = 8

. We then substitute this result into the outer function, giving

f(8) = \frac{18}{8 - 2} = \frac{18}{6} = 3

.

Step-by-Step Solution

1

Evaluate the inner function

g(3)

8

For any composite function of the form

f(g(x))

, the inner function

g(x)

must be evaluated first at the given input value.

2

Evaluate the outer function

f(8)

3

Substitute the output value of the inner function as the input for the outer function

f(x)

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 0s

Question 139Question →

If the polynomial $x^4 - 13x^2 + 36$ is factored completely into the product of four linear binomials of the form $(x - r_1)(x - r_2)(x - r_3)(x - r_4)$ , where $r_1 < r_2 < r_3 < r_4$ , what is the value of the expression $r_1 + 2r_2 + 3r_3 + 4r_4$ ?

Show answer & explanation

Answer: 11

Answer

The value of the expression is 11.

Factoring the quartic polynomial

x^4 - 13x^2 + 36

as a quadratic in

x^2

yields

(x^2 - 9)(x^2 - 4)

. Applying the difference of squares identity to each factor gives

(x - 3)(x + 3)(x - 2)(x + 2)

. Matching these to the template

(x - r_1)(x - r_2)(x - r_3)(x - r_4)

with the condition

r_1 < r_2 < r_3 < r_4

yields

r_1 = -3

,

r_2 = -2

,

r_3 = 2

, and

r_4 = 3

. Evaluating the linear combination

-3 + 2(-2) + 3(2) + 4(3)

results in

11

.

Step-by-Step Solution

1

Substitute

u = x^2

and factor the quadratic trinomial.

u^2 - 13u + 36 = (u - 9)(u - 4) \Rightarrow (x^2 - 9)(x^2 - 4)

To reduce the degree of the polynomial and make it easier to factor.

2

Apply the difference of squares formula to each binomial factor.

(x - 3)(x + 3)(x - 2)(x + 2)

Both

x^2 - 9

and

x^2 - 4

are differences of squares.

3

Determine the roots

r_1, r_2, r_3, r_4

in ascending order.

r_1 = -3

,

r_2 = -2

,

r_3 = 2

,

r_4 = 3

We write each factor as

(x - r_i)

to find the roots, and then sort them from least to greatest according to the inequality constraint.

4

Calculate the value of the requested expression.

11

Substitute the sorted values into the linear combination:

(-3) + 2(-2) + 3(2) + 4(3) = -3 - 4 + 6 + 12 = 11

.

Key Concept

Factoring a quartic polynomial of quadratic form followed by factoring differences of squares.

Question 140Question →

If the equation $2x(x - 1) = 3 - x$ is solved for $x$ , what is the larger of the two solutions?

Show answer & explanation

Answer: 1.5

Answer

The larger of the two solutions is

1.5

.

Expanding the equation

2x(x - 1) = 3 - x

results in

2x^2 - 2x = 3 - x

. Setting this quadratic equation to zero yields

2x^2 - x - 3 = 0

. Factoring the expression gives

(2x - 3)(x + 1) = 0

. Setting the first factor to zero,

2x - 3 = 0

, results in

x = 1.5

. Setting the second factor to zero,

x + 1 = 0

, results in

x = -1

. Comparing the two values,

1.5

is the larger solution.

Step-by-Step Solution

1

Expand and rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

2x^2 - x - 3 = 0

A quadratic equation must be set to zero before solving it by factoring.

2

Factor the quadratic trinomial by grouping.

(2x - 3)(x + 1) = 0

Factoring allows us to apply the zero product property.

3

Set each factor to zero to solve for the roots.

x = 1.5

and

x = -1

If a product equals zero, at least one of its factors must be zero.

4

Compare the solutions to find the larger value.

1.5

Comparing

1.5

and

-1

shows that

1.5

is the greater value.

Key Concept

Solving quadratic equations by factoring when the leading coefficient is greater than 1

Answer

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Key Concept

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