All practice questions

Question 141Question →

A community theater sells adult tickets for $\$15$ each and student tickets for $\$10$ each. The number of tickets sold for the evening performances on Friday and Saturday is represented by the matrix $A$ :

A = \begin{bmatrix} 80 & 45 \\ 110 & 60 \end{bmatrix}

where the rows represent Friday and Saturday, respectively, and the columns represent adult and student tickets, respectively.

The ticket prices are represented by the matrix $B$ :

B = \begin{bmatrix} 15 \\ 10 \end{bmatrix}

The product matrix $C = AB$ represents the total ticket revenue for each night. What is the element in the second row and first column of matrix $C$ , which represents the total ticket revenue, in dollars, for Saturday's performance?

Show answer & explanation

Answer: 2250

Answer

The correct answer is

2250

dollars, which represents the ticket revenue for Saturday's performance.

The correct answer is

2250

dollars. The element in the second row and first column of the product matrix

C = AB

is obtained by multiplying the elements of the second row of matrix

A

(Saturday's ticket sales:

110

adult,

60

student) by the corresponding elements of the first column of matrix

B

(ticket prices:

\$15

adult,

\$10

student) and summing the products:

110 \times 15 + 60 \times 10 = 1650 + 600 = 2250

.

Step-by-Step Solution

1

Identify the row of matrix

A

representing Saturday's sales and the column of matrix

B

representing the prices.

The second row of matrix

A

is

\begin{bmatrix} 110 & 60 \end{bmatrix}

(representing

110

adult tickets and

60

student tickets sold on Saturday). Matrix

B

is

\begin{bmatrix} 15 \\ 10 \end{bmatrix}

(representing the prices of

\$15

and

\$10

respectively).

Since the question asks for Saturday's performance (the second row of matrix

A

) and the revenue is calculated by multiplying sales by prices, we use these specific components for the matrix multiplication.

2

Calculate the dot product of Saturday's sales row and the price column.

110 \times 15 + 60 \times 10 = 1650 + 600 = 2250

.

This calculation determines the sum of the revenue from adult tickets (

110 \times 15 = 1650

) and student tickets (

60 \times 10 = 600

) to yield the total revenue.

Key Concept

Matrix Multiplication

Question 142Question →

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 143Question →

A system of equations consists of a parabola with the equation $y = x^2$ and a line with the equation $y = 3x - 2$ . The line and the parabola intersect at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the product of the $y$ -coordinates, $y_1 \cdot y_2$ , of these two points?

Show answer & explanation

Answer: 4

Answer

The product of the y-coordinates of the intersection points is 4.

Solving the system of equations by setting

x^2 = 3x - 2

leads to the quadratic equation

x^2 - 3x + 2 = 0

, which factors as

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

. The x-coordinates of the intersection points are

x = 1

and

x = 2

. Substituting these into

y = x^2

gives the corresponding y-coordinates as

y = 1^2 = 1

and

y = 2^2 = 4

. The product of these y-coordinates is

1 \cdot 4 = 4

.

Step-by-Step Solution

1

Equate the two equations to find the x-coordinates of the intersection points.

x^2 = 3x - 2

Since both equations are solved for y, setting them equal to each other allows us to find the x-coordinates where they intersect.

2

Move all terms to one side to set the quadratic equation to zero.

x^2 - 3x + 2 = 0

This puts the equation in the standard form

ax^2 + bx + c = 0

, which can be solved by factoring.

3

Factor the quadratic equation.

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

, so

x_1 = 1

and

x_2 = 2

Factoring shows that the values of x that satisfy the equation are the roots of the quadratic.

4

Substitute the x-coordinates back into the equation of the parabola to find the y-coordinates.

For

x_1 = 1

,

y_1 = 1^2 = 1

. For

x_2 = 2

,

y_2 = 2^2 = 4

.

Substituting x into either of the original equations gives the corresponding y-coordinates of the intersection points.

5

Calculate the product of the y-coordinates.

1 \cdot 4 = 4

The question asks for the product of the y-coordinates, so we multiply

y_1

and

y_2

.

Key Concept

Solving systems of linear and quadratic equations by substitution.

Question 144Question →

For what greatest integer value of $y$ is the inequality $9 - 2y \geq 16$ true?

Show answer & explanation

Answer: -4

Answer

The greatest integer value of

y

that satisfies the inequality is

-4

.

Subtracting 9 from both sides of

9 - 2y \geq 16

gives

-2y \geq 7

. Dividing both sides by

-2

and reversing the inequality sign yields

y \leq -3.5

. The greatest integer less than or equal to

-3.5

is

-4

.

Step-by-Step Solution

1

Subtract 9 from both sides of the inequality.

-2y \geq 7

This isolates the term containing

y

on the left side.

2

Divide both sides of the inequality by

-2

and reverse the direction of the inequality sign.

y \leq -3.5

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

3

Identify the greatest integer that is less than or equal to

-3.5

.

-4

The value of

y

must be less than or equal to

-3.5

. The integers satisfying this condition are

-4, -5, -6, \dots

, and the greatest of these is

-4

.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.

Question 145Question →

Matrices $A$ , $B$ , and $C$ are defined such that $A = \begin{bmatrix} x & 3 \\ -2 & 4 \end{bmatrix}$ , $B = \begin{bmatrix} 2 & -1 \\ y & 5 \end{bmatrix}$ , and $C = \begin{bmatrix} 14 & 11 \\ 4 & 22 \end{bmatrix}$ . If $AB = C$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x + y

is 6.

The correct answer is 6 because performing the matrix multiplication

AB

yields the matrix

\begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

. Equating this to matrix

C

gives the equations

-x + 15 = 11

, which yields

x = 4

, and

-4 + 4y = 4

, which yields

y = 2

. The sum of these values is

4 + 2 = 6

. This is verified by checking the remaining equation

2(4) + 3(2) = 14

.

Step-by-Step Solution

1

Multiply matrix

A

and matrix

B

to express the product

AB

algebraically.

AB = \begin{bmatrix} x(2) + 3(y) & x(-1) + 3(5) \\ -2(2) + 4(y) & -2(-1) + 4(5) \end{bmatrix} = \begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

To solve for the variables, we must first determine the resulting matrix from the multiplication of

A

and

B

.

2

Set up equations by equating the entries of

AB

with the corresponding entries of matrix

C

.

2x + 3y = 14

,

-x + 15 = 11

, and

-4 + 4y = 4

Two matrices are equal if and only if all of their corresponding elements are equal.

3

Solve the single-variable equation from the first row, second column for

x

.

-x + 15 = 11 \implies -x = -4 \implies x = 4

Isolating

x

gives its numeric value.

4

Solve the single-variable equation from the second row, first column for

y

.

-4 + 4y = 4 \implies 4y = 8 \implies y = 2

Isolating

y

gives its numeric value.

5

Verify consistency with the remaining equation and calculate the final sum

x + y

.

2(4) + 3(2) = 8 + 6 = 14

(consistent), and

x + y = 4 + 2 = 6

.

Checking the first row, first column confirms the values are correct, and adding them yields the requested sum.

Key Concept

Matrix multiplication and matrix equality

Question 146Question →

An agricultural cooperative packages a premium seed mixture containing rye grass, fescue, and bluegrass. The weight of the fescue in the mixture is $10$ pounds less than twice the weight of the rye grass. The weight of the bluegrass is $15$ pounds more than half the weight of the fescue. If the total weight of the mixture is $120$ pounds, how many pounds of bluegrass are in the mixture?

Show answer & explanation

Answer: 40

Answer

The weight of the bluegrass in the mixture is 40 pounds.

By translating the given relationships into equations (

f = 2r - 10

,

b = \frac{1}{2}f + 15

, and

r + f + b = 120

), we can express all variables in terms of

r

, yielding

r + (2r - 10) + (r + 10) = 120

. Solving this gives

r = 30

. Substituting this back gives the weight of bluegrass as

30 + 10 = 40

pounds.

Step-by-Step Solution

1

Define variables for each type of grass in the mixture.

Let

r

represent the weight of rye grass,

f

represent the weight of fescue, and

b

represent the weight of bluegrass.

Establishing variables is necessary to translate the verbal descriptions into algebraic terms.

2

Translate the given relationships into equations.

f = 2r - 10

and

b = \frac{1}{2}f + 15

The problem states the fescue is 10 pounds less than twice the rye grass, and the bluegrass is 15 pounds more than half the fescue.

3

Substitute the expression for f into the equation for b to express b solely in terms of r.

b = \frac{1}{2}(2r - 10) + 15 = r - 5 + 15 = r + 10

Reducing the number of variables simplifies the system of equations.

4

Set up the total weight equation and solve for r.

r + (2r - 10) + (r + 10) = 120 \implies 4r = 120 \implies r = 30

The sum of the three grass weights is given as 120 pounds.

5

Calculate the weight of the bluegrass using the value of r.

b = 30 + 10 = 40

The question asks specifically for the weight of the bluegrass.

Key Concept

Translating verbal relationships into linear equations and solving a system of equations

Question 147Question →

A manufacturing company determines that its weekly profit from producing $x$ batches of a product is constrained by resource availability. To meet these resource constraints, the number of batches $x$ must satisfy the inequality:

\frac{5(6 - x)}{3} - \frac{3(x + 2)}{4} > 2x - 11

What is the greatest number of whole batches the company can produce while satisfying this constraint?

Show answer & explanation

Answer: 4

Answer

The greatest number of whole batches the company can produce is 4.

Solving the inequality yields

x < \frac{234}{53} \approx 4.415

. The greatest integer value that satisfies this condition is 4.

Step-by-Step Solution

1

Multiply both sides of the inequality by the least common multiple of the denominators, which is 12.

20(6 - x) - 9(x + 2) > 24x - 132

Multiplying by 12 eliminates the fractions, making the inequality easier to solve.

2

Distribute the constants on the left side of the inequality.

120 - 20x - 9x - 18 > 24x - 132

Distributing 20 to

(6 - x)

yields

120 - 20x

, and distributing

-9

to

(x + 2)

yields

-9x - 18

.

3

Combine like terms on the left side of the inequality.

102 - 29x > 24x - 132

Combining

120 - 18

gives

102

, and combining

-20x - 9x

gives

-29x

.

4

Subtract

24x

from both sides to group the variable terms on the left side.

102 - 53x > -132

This groups all terms containing the variable

x

on one side of the inequality.

5

Subtract 102 from both sides to isolate the variable term.

-53x > -234

This isolates the term containing

x

on the left side of the inequality.

6

Divide both sides by

-53

and reverse the inequality sign.

x < \frac{234}{53}

Dividing by a negative number requires reversing the direction of the inequality sign.

7

Evaluate the fraction as a decimal and determine the greatest integer value of

x

that satisfies the inequality.

x < 4.415

, which means the greatest integer is 4.

Since the company must produce a whole number of batches, we find the largest integer less than 4.415.

Key Concept

Solving multi-step linear inequalities involving fractional coefficients, distributing negative numbers, and reversing the inequality sign when multiplying or dividing by a negative number.

Alternative Method

Instead of solving algebraically, you can test integer values for

x

directly in the inequality. Testing

x = 4

gives

\frac{10}{3} - \frac{18}{4} = 3.33 - 4.5 = -1.17

, which is greater than

2(4) - 11 = -3

(True). Testing

x = 5

gives

\frac{5}{3} - \frac{21}{4} = 1.67 - 5.25 = -3.58

, which is not greater than

2(5) - 11 = -1

(False). This confirms 4 is the largest integer satisfying the inequality.

Estimated Time:2m 30s

Question 148Question →

When the expression $4x(x - 2y) - (2x - 3y)^2 + 5y(2x - y)$ is simplified to the form $Ax^2 + Bxy + Cy^2$ , where $A$ , $B$ , and $C$ are constants, what is the value of $B$ ?

Show answer & explanation

Answer: 14

Answer

The value of the coefficient

B

is 14.

Expanding the entire expression yields

4x^2 - 8xy - 4x^2 + 12xy - 9y^2 + 10xy - 5y^2

. Grouping and combining the

xy

terms gives

(-8 + 12 + 10)xy = 14xy

. Therefore, the coefficient

B

is 14.

Step-by-Step Solution

1

Expand the first term

4x^2 - 8xy

Distribute

4x

to both terms inside the parentheses:

4x(x) - 4x(2y) = 4x^2 - 8xy

.

2

Expand the squared binomial and apply the negative sign

-4x^2 + 12xy - 9y^2

Use the binomial expansion formula

(2x - 3y)^2 = 4x^2 - 12xy + 9y^2

, then multiply each term by

-1

.

3

Expand the third term

10xy - 5y^2

Distribute

5y

to both terms inside the parentheses:

5y(2x) - 5y(y) = 10xy - 5y^2

.

4

Combine the coefficients of the like terms

0x^2 + 14xy - 14y^2

Sum the coefficients for each corresponding variable group:

(4 - 4)x^2 + (-8 + 12 + 10)xy + (-9 - 5)y^2

.

Key Concept

Simplifying Algebraic Expressions and Combining Like Terms

Question 149Question →

In the standard $(x, y)$ coordinate plane, the line defined by the equation $3x - 4y = k$ is tangent to the circle defined by the equation $x^2 + y^2 - 2x - 4y = 4$ . If $k > 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. Completing the square for the circle's equation gives

(x - 1)^2 + (y - 2)^2 = 9

, showing the center is

(1, 2)

and the radius is

3

. The distance from

(1, 2)

to the line

3x - 4y - k = 0

is

\frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|k + 5|}{5}

. For tangency, this distance must equal the radius:

\frac{|k + 5|}{5} = 3

, which gives

|k + 5| = 15

. Solving this absolute value equation gives

k = 10

or

k = -20

. Since

k

must be positive, the value is 10.

Step-by-Step Solution

1

Complete the square for the circle's equation

x^2 + y^2 - 2x - 4y = 4

.

(x - 1)^2 + (y - 2)^2 = 9

, which represents a circle with center

(1, 2)

and radius

R = 3

.

To find the center and radius of the circle, which are needed to use the distance formula.

2

Express the distance

d

from the center

(1, 2)

to the line

3x - 4y - k = 0

using the formula

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

.

d = \frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|-5 - k|}{5} = \frac{|k + 5|}{5}

.

A line is tangent to a circle if and only if the distance from the center of the circle to the line equals the radius.

3

Set the distance equal to the radius (

3

) and solve for

k

.

\frac{|k + 5|}{5} = 3 \Rightarrow |k + 5| = 15

, which yields

k + 5 = 15 \Rightarrow k = 10

, or

k + 5 = -15 \Rightarrow k = -20

.

To find the values of

k

that make the line tangent to the circle.

4

Select the positive value of

k

.

k = 10

.

The problem specifies that

k > 0

.

Key Concept

Systems of Linear and Non-Linear Equations

Estimated Time:2m 30s

Question 150Question →

Consider the function $f$ defined on the domain $[\frac{2}{3}, \infty)$ by the equation $f(x) = 2 + \sqrt{3x - 2}$ . If $f^{-1}$ represents the inverse of $f$ , what is the only real value of $x$ for which $f(x) = f^{-1}(x)$ ?

Show answer & explanation

Answer: 6

Answer

The only real value of

x

for which

f(x) = f^{-1}(x)

is 6.

For a strictly increasing function, the intersection of

f(x)

and

f^{-1}(x)

must occur on the line

y = x

. Equating

f(x) = x

gives

2 + \sqrt{3x - 2} = x

. Isolating the radical term yields

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

. Squaring both sides produces

3x - 2 = x^2 - 4x + 4

, which simplifies to the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives

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

, yielding the solutions

x = 6

and

x = 1

. Substituting these back into the original equation shows that

x = 6

is a valid solution (

2 + \sqrt{16} = 6

), whereas

x = 1

is extraneous because it results in

2 + 1 = 3 \neq 1

. Furthermore, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, meaning

f^{-1}(1)

is undefined. Thus, the only real solution is 6.

Step-by-Step Solution

1

Equate the function to

x

using properties of increasing functions and their inverses.

Since

f(x)

is strictly increasing on its domain

[\frac{2}{3}, \infty)

, the graph of

f(x)

and the graph of its inverse

f^{-1}(x)

can only intersect on the line of symmetry

y = x

. Therefore, the equation

f(x) = f^{-1}(x)

is equivalent to

f(x) = x

.

This simplifies the relation by eliminating the need to solve a high-degree polynomial equation derived from direct composition or substitution.

2

Set up the equation

f(x) = x

and isolate the radical term.

2 + \sqrt{3x - 2} = x \implies \sqrt{3x - 2} = x - 2

Isolating the square root term is a necessary prerequisite step before squaring both sides.

3

Square both sides and rewrite the equation as a standard quadratic equation.

3x - 2 = (x - 2)^2 \implies 3x - 2 = x^2 - 4x + 4 \implies x^2 - 7x + 6 = 0

Squaring eliminates the radical to yield a standard quadratic equation that can be solved analytically.

4

Solve the quadratic equation by factoring.

(x - 6)(x - 1) = 0 \implies x = 6 \text{ or } x = 1

Factoring the quadratic trinomial yields the two candidate values for

x

.

5

Check the candidate solutions in the original equation to eliminate extraneous roots.

For

x = 1

:

2 + \sqrt{3(1)-2} = 3 \neq 1

(extraneous). For

x = 6

:

2 + \sqrt{3(6)-2} = 2 + 4 = 6

(valid). Also, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, thus excluding

x = 1

.

Squaring both sides can introduce extraneous roots. We must verify that the solutions satisfy the original radical equation and fall within the domains of both

f

and

f^{-1}

.

Key Concept

Applying the symmetry of inverse functions about the line

y=x

to solve composition-based equations, while rigorously accounting for domain restrictions and extraneous roots.

Alternative Method

Find the algebraic formula for

f^{-1}(x)

by setting

y = 2 + \sqrt{3x-2}

. Subtracting 2 and squaring both sides gives

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

for

y \geq 2

. Solving for

x

yields

f^{-1}(x) = \frac{(x - 2)^2 + 2}{3}

for

x \geq 2

. Equating

f(x) = f^{-1}(x)

results in

2 + \sqrt{3x - 2} = \frac{(x - 2)^2 + 2}{3}

. Multiplying by 3 and isolating the radical term gives

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

. Squaring both sides results in a fourth-degree polynomial equation:

9(3x - 2) = (x^2 - 4x)^2 \implies x^4 - 8x^3 + 16x^2 - 27x + 18 = 0

. This factors into

(x - 6)(x - 1)(x^2 - x + 3) = 0

. Since the domain of

f^{-1}(x)

is restricted to

x \geq 2

, the root

x = 1

is rejected, and the quadratic factor

x^2 - x + 3 = 0

has no real roots, leaving

x = 6

as the unique real solution.

Estimated Time:3m 0s

Question 151Question →

Let matrix $X = \begin{bmatrix} 5 & -3 \\ 2 & 1 \end{bmatrix}$ and matrix $Y = \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}$ . If $Z = X - Y$ , what is the value of the element in the first row and second column of $Z$ ?

Show answer & explanation

Answer: -7

Answer

The correct answer is

-7

.

To find the element in the first row and second column of matrix

Z = X - Y

, we subtract the element in the first row and second column of matrix

Y

from the corresponding element in matrix

X

. The element in the first row and second column of

X

is

-3

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

Y

is

4

. Subtracting these values gives

-3 - 4 = -7

.

Step-by-Step Solution

1

Identify the elements in the first row and second column for both matrices.

x_{1,2} = -3

and

y_{1,2} = 4

To find the element in the first row and second column of the resulting matrix

Z

, we must use the corresponding elements from matrices

X

and

Y

.

2

Subtract the element of

Y

from the element of

X

.

-3 - 4 = -7

Since

Z = X - Y

, each element

z_{i,j}

of the resulting matrix is calculated as

x_{i,j} - y_{i,j}

.

Key Concept

Matrix Subtraction

Question 152Question →

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 153Question →

The polynomial $P(x)$ is defined by $P(x) = (2x^2 - 3x + 5)^2 - 4x(x^3 - 2x^2 - 7x + 1)$ . When $P(x)$ is written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 57

Answer

The coefficient of the

x^2

term is 57.

Expanding

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

yields

(2x^2 - 3x + 5)(2x^2 - 3x + 5) = 4x^4 - 12x^3 + 29x^2 - 30x + 25

. Distributing the

-4x

term yields

-4x(x^3 - 2x^2 - 7x + 1) = -4x^4 + 8x^3 + 28x^2 - 4x

. Combining the

x^2

terms from both expressions gives

29x^2 + 28x^2 = 57x^2

. Thus, the coefficient of the

x^2

term is 57.

Step-by-Step Solution

1

Expand the squared trinomial

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

4x^4 - 12x^3 + 29x^2 - 30x + 25

Expanding the first part of the expression by multiplying the trinomial by itself.

2

Distribute the term

-4x

to the trinomial

(x^3 - 2x^2 - 7x + 1)

-4x^4 + 8x^3 + 28x^2 - 4x

Expanding the second part of the polynomial expression while distributing the negative sign to all terms.

3

Combine the expanded expressions and isolate the

x^2

terms

29x^2 + 28x^2 = 57x^2

Adding the coefficients of the terms of degree 2 to find the combined coefficient.

Key Concept

Operations on Polynomials

Question 154Question →

If the quadratic expression $6x^2 - 7x - 5$ is factored completely into the product of two linear binomials of the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the expression $ad - bc$ ?

Show answer & explanation

Answer: 13

Answer

The value of the expression

ad - bc

is 13.

Step-by-Step Solution

1

Factor the quadratic expression

6x^2 - 7x - 5

.

(3x - 5)(2x + 1)

Find two numbers that multiply to

6 \times (-5) = -30

and add to

-7

, which are

-10

and

3

. Rewrite the middle term and factor by grouping:

6x^2 - 10x + 3x - 5 = 2x(3x - 5) + 1(3x - 5) = (3x - 5)(2x + 1)

.

2

Determine the values of the coefficients

a

,

b

,

c

, and

d

.

a = 3

,

b = -5

,

c = 2

, and

d = 1

The expression is factored into the form

(ax + b)(cx + d)

where

a > c > 0

. Comparing the factors

(3x - 5)

and

(2x + 1)

, we see the coefficients of

x

are

3

and

2

. Since

3 > 2 > 0

, we have

a = 3

and

c = 2

. This leaves

b = -5

and

d = 1

.

3

Calculate the value of

ad - bc

.

13

Substitute

a = 3

,

b = -5

,

c = 2

, and

d = 1

into the expression:

ad - bc = (3)(1) - (-5)(2) = 3 + 10 = 13

.

Key Concept

Factoring quadratic polynomials of the form

ax^2 + bx + c

with

a > 1

.

Question 155Question →

What is the value of the larger real solution to the equation $\frac{10}{x} - \frac{3}{x-1} = 2$ ?

Show answer & explanation

Answer: 2.5

Answer

The larger real solution is

2.5

.

Multiplying the equation by the least common denominator

x(x-1)

yields

10(x-1) - 3x = 2x(x-1)

. Simplifying this leads to

2x^2 - 9x + 10 = 0

. Factoring the quadratic yields

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

, which gives the solutions

x = 2.5

and

x = 2

. The larger of these two solutions is

2.5

.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator, which is

x(x-1)

, for

x \neq 0

and

x \neq 1

.

10(x-1) - 3x = 2x(x-1)

This clears the fractions from the rational equation.

2

Expand both sides of the equation.

10x - 10 - 3x = 2x^2 - 2x

Expanding the terms allows us to combine like terms.

3

Combine like terms on the left side and move all terms to one side to set the quadratic equation equal to zero.

2x^2 - 9x + 10 = 0

Setting the quadratic equation to zero is required to solve it by factoring.

4

Factor the quadratic equation by grouping.

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

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

5

Solve for

x

by setting each factor equal to zero.

x = 2.5

or

x = 2

By the zero product property, at least one of the factors must be zero.

6

Check for extraneous solutions and select the larger real value.

Both

2

and

2.5

are valid because they do not make the original denominators zero. The larger value is

2.5

.

The question specifically asks for the larger of the two real solutions.

Key Concept

Solving rational equations by clearing denominators

Estimated Time:1m 30s

Question 156Question →

If

x

and

y

are positive real numbers such that

(x^3 y^{-2})^k \cdot (x^{-1} y^4)^3 = x^6 y^6

what is the value of the exponent

k

?

Show answer & explanation

Answer: 3

Answer

The value of the exponent

k

is 3.

Applying the exponent rules simplifies the left side of the equation to

x^{3k-3}y^{12-2k}

. Equating the exponent of

x

to the right side gives

3k - 3 = 6

, which yields

k = 3

. This value is confirmed by equating the exponent of

y

, since

12 - 2(3) = 6

.

Step-by-Step Solution

1

Apply the power of a power rule

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

to expand the terms in the expression.

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

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To remove the outer parentheses by multiplying the internal exponents of each variable by the outer exponent.

2

Multiply the terms together by applying the product rule for exponents,

a^m \cdot a^n = a^{m+n}

.

(x^{3k} y^{-2k})(x^{-3} y^{12}) = x^{3k-3} y^{12-2k}

To combine the like bases of

x

and

y

into a single simplified expression.

3

Set the exponents of like bases equal to the exponents on the right-hand side of the equation,

x^6 y^6

.

3k - 3 = 6

and

12 - 2k = 6

Since the bases are equal and non-zero, their respective exponents must also be equal.

4

Solve the linear equation

3k - 3 = 6

for

k

.

3k = 9 \implies k = 3

To determine the numerical value of the variable

k

.

5

Verify the solution by solving the second linear equation,

12 - 2k = 6

.

-2k = -6 \implies k = 3

To ensure consistency across both variable exponents in the expression.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 157Question →

Let the function $h$ be defined by $h(x) = 3x - 5$ , and let the function $g$ be defined by $g(x) = x^2 + c$ for some constant $c$ . If $h(g(2)) = 10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 1

Answer

The value of the constant

c

is

1

.

The correct answer is

1

. Evaluating the inner function

g(2)

gives

2^2 + c = 4 + c

. Applying the outer function

h(x)

to this expression gives

h(4 + c) = 3(4 + c) - 5

. Simplifying this expression results in

12 + 3c - 5 = 7 + 3c

. We are given that this value equals

10

, so

7 + 3c = 10

. Subtracting 7 from both sides gives

3c = 3

, which yields

c = 1

.

Step-by-Step Solution

1

Evaluate

g(2)

in terms of

c

g(2) = 4 + c

We substitute

x = 2

into the definition

g(x) = x^2 + c

to get

2^2 + c = 4 + c

.

2

Substitute

g(2)

into the definition of

h(x)

to express

h(g(2))

h(g(2)) = 3(4 + c) - 5

Since the composition is

h(g(2))

, we evaluate the function

h

at the input value

g(2) = 4 + c

.

3

Set

h(g(2)) = 10

and solve the linear equation for

c

c = 1

Expanding and simplifying the equation

3(4 + c) - 5 = 10

gives

12 + 3c - 5 = 10

, which reduces to

7 + 3c = 10

. Subtracting 7 gives

3c = 3

, so

c = 1

.

Key Concept

Function Composition and Parameter Evaluation

Estimated Time:1m 15s

Question 158Question →

What is the smallest integer value of $x$ that satisfies the inequality $11 - 3x < 2$ ?

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

x

that satisfies the inequality is 4.

Solving the inequality

11 - 3x < 2

leads to

-3x < -9

. Dividing by

-3

and reversing the inequality sign gives

x > 3

. The smallest integer that is strictly greater than 3 is 4.

Step-by-Step Solution

1

Isolate the variable term on one side of the inequality by subtracting 11 from both sides.

-3x < -9

Subtracting 11 from both sides of the inequality keeps the relationship balanced.

2

Divide both sides of the inequality by the coefficient of

x

, which is

-3

.

x > 3

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

3

Determine the smallest integer that is strictly greater than 3.

4

Since the inequality is strict (

x > 3

), 3 is not included in the solution set. The smallest integer greater than 3 is 4.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.

Question 159Question →

The matrices $M$ and $N$ are defined as follows:

M = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}, \quad N = \begin{bmatrix} 2 & 5 \\ 0 & -1 \end{bmatrix}

If the product matrix $P$ is defined by the equation $P = MN$ , what is the value of the element in the first row and second column of $P$ ?

Show answer & explanation

Answer: 22

Answer

The value of the element in the first row and second column of

P

is 22.

To find the element in the first row and second column of the product matrix

P = MN

, we calculate the dot product of the first row of matrix

M

(

[4, -2]

) and the second column of matrix

N

(

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

):

4(5) + (-2)(-1) = 20 + 2 = 22

.

Step-by-Step Solution

1

Determine the row of the left matrix and the column of the right matrix needed for the target element.

To find the element in row 1, column 2 of

P = MN

, use the first row of

M

, which is

[4, -2]

, and the second column of

N

, which is

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

.

The entry in row

i

and column

j

of a product matrix is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Compute the dot product of the selected row and column.

(4 \times 5) + (-2 \times -1) = 20 + 2

Multiply corresponding elements and sum the products.

3

Simplify to get the final numeric answer.

22

Adding 20 and 2 yields 22.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix.

Question 160Question →

A triangle has a base of $2x + 4$ inches and a height of $x - 3$ inches. What is the coefficient of $x$ when the expression representing the area of the triangle, in square inches, is written in standard form?

Show answer & explanation

Answer: -1

Answer

The coefficient of

x

is

-1

.

The area of a triangle is calculated using

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Substituting the given expressions, we get

\text{Area} = \frac{1}{2}(2x + 4)(x - 3)

. First, we can multiply the

\frac{1}{2}

by

(2x + 4)

, which simplifies to

x + 2

. Next, we expand

(x + 2)(x - 3)

using FOIL to get

x^2 - 3x + 2x - 6

. Combining like terms yields

x^2 - x - 6

. The coefficient of the

x

term in this simplified expression is

-1

.

Step-by-Step Solution

1

State the formula for the area of a triangle.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

To establish the mathematical relationship.

2

Substitute the given values into the formula.

\text{Area} = \frac{1}{2}(2x + 4)(x - 3)

To express the area in terms of the variable

x

.

3

Distribute the fraction to the first binomial.

x + 2

Distributing

\frac{1}{2}

to

(2x + 4)

simplifies the expression before multiplying.

4

Expand the product of the binomials.

x^2 - 3x + 2x - 6

Using the FOIL method to multiply

(x + 2)

and

(x - 3)

.

5

Combine the linear terms.

x^2 - x - 6

To simplify the polynomial and write it in standard form.

6

Identify the coefficient of

x

.

-1

The coefficient of the

x

term in

x^2 - x - 6

is

-1

.

Key Concept

Multiplying binomials and applying formulas in geometric contexts

Estimated Time:1m 0s

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