All practice questions

Question 1961Question →

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 1962Question →

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 1963Question →

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

Question 1964Question →

Consider the system of equations below:

\begin{aligned} x^2 + y^2 &= 9 \\ y &= x - 3 \end{aligned}

If $(x, y)$ is a solution to the system such that $x > 0$ , what is the value of $x + y$ ?

3

-3

0

6

9

Show answer & explanation

Answer: 3

Answer

3

The correct answer is

3

. Substituting

y = x - 3

into

x^2 + y^2 = 9

gives

x^2 + (x - 3)^2 = 9

. Expanding this yields

2x^2 - 6x = 0

, which factors as

2x(x - 3) = 0

. Since

x > 0

, the

x

-coordinate must be

3

. Substituting

x = 3

back into the linear equation gives

y = 3 - 3 = 0

. Therefore,

x + y = 3 + 0 = 3

.

Step-by-Step Solution

1

Substitute the linear equation into the quadratic equation.

x^2 + (x - 3)^2 = 9

This reduces the system of two variables to a single quadratic equation in terms of

x

.

2

Expand the squared binomial and simplify the equation.

x^2 + (x^2 - 6x + 9) = 9 \implies 2x^2 - 6x = 0

Expanding

(x - 3)^2

yields

x^2 - 6x + 9

. Subtracting

9

from both sides simplifies the equation.

3

Factor the quadratic equation to solve for

x

.

2x(x - 3) = 0 \implies x = 0

or

x = 3

Factoring out the greatest common factor

2x

reveals the roots of the quadratic equation.

4

Apply the constraint

x > 0

to find the correct

x

-coordinate.

x = 3

The problem specifies that the solution

(x, y)

must have

x > 0

.

5

Find the corresponding

y

-coordinate and calculate

x + y

.

y = 3 - 3 = 0 \implies x + y = 3 + 0 = 3

Substituting

x = 3

into the linear equation gives

y = 0

, and summing them gives the required value.

Key Concept

Solving systems of linear and quadratic equations by substitution

Question 1965Question →

When the expression $3x^2(2x - y) - 2y(x^2 - 3xy) - (4x^3 - xy^2)$ is simplified to the form $Ax^3 + Bx^2y + Cxy^2$ , where $A$ , $B$ , and $C$ are integers, what is the value of $A + B + C$ ?

-8

-2

1

2

4

Show answer & explanation

Answer: 4

Answer

4

The correct sum of the coefficients is 4. By distributing all terms correctly:

3x^2(2x - y) = 6x^3 - 3x^2y

,

-2y(x^2 - 3xy) = -2x^2y + 6xy^2

, and

-(4x^3 - xy^2) = -4x^3 + xy^2

. Combining the terms yields

2x^3 - 5x^2y + 7xy^2

, which gives

A = 2

,

B = -5

, and

C = 7

. Summing these values gives

2 - 5 + 7 = 4

.

Step-by-Step Solution

1

Distribute

3x^2

to the first parenthetical expression:

3x^2(2x - y)

6x^3 - 3x^2y

Applying the distributive property multiplies

3x^2

by both

2x

and

-y

.

2

Distribute

-2y

to the second parenthetical expression:

-2y(x^2 - 3xy)

-2x^2y + 6xy^2

Applying the distributive property multiplies

-2y

by both

x^2

and

-3xy

, noting that a negative times a negative is a positive.

3

Distribute the negative sign to the third parenthetical expression:

-(4x^3 - xy^2)

-4x^3 + xy^2

Distributing the negative sign changes the signs of both terms inside the parenthesis.

4

Combine the expanded parts:

(6x^3 - 3x^2y) + (-2x^2y + 6xy^2) + (-4x^3 + xy^2)

and group like terms

(6x^3 - 4x^3) + (-3x^2y - 2x^2y) + (6xy^2 + xy^2)

Grouping like terms together makes it easier to combine their coefficients.

5

Combine the coefficients of the like terms

2x^3 - 5x^2y + 7xy^2

Combining the coefficients gives

A = 2

,

B = -5

, and

C = 7

.

6

Calculate the sum

A + B + C

2 + (-5) + 7 = 4

Adding the coefficients together yields the final numerical value.

Key Concept

Simplifying algebraic expressions by distributing coefficients (including negative signs) and combining like terms.

Question 1966Question →

A square metal sheet has a side length of $x + 4$ inches. A rectangular region with a width of $x$ inches and a length of $x + 2$ inches is removed from the sheet. Which of the following expressions represents the area, in square inches, of the remaining portion of the sheet?

-2x + 16

8x + 14

10x + 16

6x + 16

6x + 8

Show answer & explanation

Answer: 6x + 16

Answer

The expression that represents the remaining area is

6x + 16

.

To find the remaining area, subtract the area of the removed rectangle from the area of the original square sheet. The area of the square is

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

and the area of the rectangle is

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

. Subtracting these yields

(x^2 + 8x + 16) - (x^2 + 2x) = x^2 + 8x + 16 - x^2 - 2x = 6x + 16

.

Step-by-Step Solution

1

Find the area of the original square metal sheet.

Area of the square is

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

square inches.

The area of a square is calculated by squaring its side length.

2

Find the area of the removed rectangular region.

Area of the rectangle is

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

square inches.

The area of a rectangle is calculated by multiplying its width by its length.

3

Subtract the area of the rectangular region from the area of the square sheet.

(x^2 + 8x + 16) - (x^2 + 2x) = x^2 + 8x + 16 - x^2 - 2x = 6x + 16

square inches.

Subtracting the removed area from the total area gives the remaining area, distributing the negative sign to all terms inside the parentheses.

Key Concept

Operations on Polynomials

Estimated Time:45s

Question 1967Question →

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 1968Question →

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 1969Question →

A school club sold 20 rolls of wrapping paper and 25 boxes of greeting cards for a fundraiser. The price of a roll of wrapping paper was $\frac{3}{4}$ of the price of a box of greeting cards. If the club raised a total of $800 from these sales, what was the price of a single roll of wrapping paper?

$5.00

$20.00

$15.00

$20.65

$26.67

Show answer & explanation

Answer: $15.00

Answer

The price of a single roll of wrapping paper was $15.00.

The correct answer is

15.00. By representing the price of a roll of wrapping paper as

W

and the price of a box of greeting cards as

G

, the problem states that

W = \frac{3}{4}G

, which means

G = \frac{4}{3}W

. The total revenue equation is

20W + 25G = 800

. Substituting

G

gives

20W + 25\left(\frac{4}{3}W\right) = 800

, which simplifies to

20W + \frac{100}{3}W = 800

. Multiplying the entire equation by 3 to clear the fraction results in

60W + 100W = 2400

. Combining like terms gives

160W = 2400

, which yields

W = 15$.

Step-by-Step Solution

1

Define variables for the unknowns and translate the price relationship into an algebraic equation.

Let

W

be the price of a roll of wrapping paper and

G

be the price of a box of greeting cards. The relationship is given by

W = \frac{3}{4}G

, which can be rearranged to express

G

in terms of

W

:

G = \frac{4}{3}W

.

Expressing one variable in terms of another allows us to set up a single-variable linear equation.

2

Write the linear equation representing the total revenue from the fundraiser sales.

The total revenue from selling 20 rolls of wrapping paper and 25 boxes of greeting cards is

20W + 25G = 800

. Substituting

G = \frac{4}{3}W

gives:

20W + 25\left(\frac{4}{3}W\right) = 800

, which simplifies to

20W + \frac{100}{3}W = 800

.

This sets up the equation that we need to solve to find the value of

W

.

3

Clear the fraction by multiplying all terms by 3 and solve for

W

.

Multiplying the entire equation by 3 yields:

3(20W) + 3\left(\frac{100}{3}W\right) = 3(800) \implies 60W + 100W = 2400

. Combining like terms gives

160W = 2400

. Dividing by 160 yields

W = 15

.

Clearing the denominator simplifies the equation to a standard linear form that can be solved directly.

Key Concept

Solving linear equations derived from real-world contexts, particularly those involving fractional relationships and multi-step isolation.

Alternative Method

Instead of expressing

G

in terms of

W

first, solve for

G

directly by substituting

W = \frac{3}{4}G

into the revenue equation. This gives

20\left(\frac{3}{4}G\right) + 25G = 800 \implies 15G + 25G = 800 \implies 40G = 800 \implies G = 20

. Then, calculate

W = \frac{3}{4}(20) = 15

.

Estimated Time:2m 30s

Question 1970Question →

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 1971Question →

A square garden plot has a side length of $3s - 2$ yards. A walkway of uniform width $s + 3$ yards is built around the entire garden. Which of the following expressions represents the area, in square yards, of the walkway?

7s^2 + 20s - 3

16s^2 + 28s + 20

16s^2 + 52s + 12

16s^2 + 12

4s^2 + 52s + 12

Show answer & explanation

Answer: 16s^2 + 52s + 12

Answer

16s^2 + 52s + 12

The correct answer is found by first calculating the outer side length, which is the inner side length plus twice the walkway width:

(3s - 2) + 2(s + 3) = 5s + 4

. Squaring this yields the outer area of

25s^2 + 40s + 16

. The inner area is

(3s - 2)^2 = 9s^2 - 12s + 4

. Subtracting the inner area from the outer area and distributing the negative sign properly gives

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 16s^2 + 52s + 12

.

Step-by-Step Solution

1

Determine the outer side length of the square including the walkway.

The outer side length is

(3s - 2) + 2(s + 3) = 3s - 2 + 2s + 6 = 5s + 4

yards.

Since the walkway surrounds the garden on all sides, the width of the walkway must be added to both ends of the garden's side length.

2

Calculate the area of the outer square and the inner square garden by squaring their respective side lengths.

Outer Area =

(5s + 4)^2 = 25s^2 + 40s + 16

and Inner Area =

(3s - 2)^2 = 9s^2 - 12s + 4

.

The area of a square is equal to the square of its side length.

3

Subtract the inner garden area from the outer area to find the walkway area, distributing the negative sign to all terms of the inner area.

Walkway Area =

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 25s^2 + 40s + 16 - 9s^2 + 12s - 4 = 16s^2 + 52s + 12

square yards.

The area of the walkway is the difference between the total outer area and the inner garden area.

Key Concept

Operations on Polynomials

Question 1972Question →

What is the set of all real numbers $y$ that make the inequality $2 - 3|1 - 2y| \ge -10$ a true statement?

-2.5 \le y \le 1.5

y \le -1.5

or

y \ge 2.5

-1.5 \le y \le 2.5

y \ge -1.5

-4.5 \le y \le 5.5

Show answer & explanation

Answer: $-1.5 \le y \le 2.5$

Answer

-1.5 \le y \le 2.5

To solve the inequality

2 - 3|1 - 2y| \ge -10

, we first isolate the absolute value term by subtracting 2 from both sides to get

-3|1 - 2y| \ge -12

, and then dividing both sides by

-3

. Since we divide by a negative number, the inequality sign reverses, giving

|1 - 2y| \le 4

. We rewrite this as the compound inequality

-4 \le 1 - 2y \le 4

. Subtracting 1 from all parts yields

-5 \le -2y \le 3

. Finally, dividing by

-2

and reversing the inequality signs gives

2.5 \ge y \ge -1.5

, which is rewritten from least to greatest as

-1.5 \le y \le 2.5

.

Step-by-Step Solution

1

Subtract 2 from both sides of the inequality to begin isolating the absolute value term.

-3|1 - 2y| \ge -12

Isolating the absolute value expression allows us to rewrite it as a standard inequality.

2

Divide both sides by

-3

and reverse the inequality sign because of division by a negative number.

|1 - 2y| \le 4

Dividing by a negative value requires reversing the inequality direction to preserve the truth of the statement.

3

Rewrite the absolute value inequality as a compound inequality.

-4 \le 1 - 2y \le 4

An inequality of the form

|u| \le c

(where

c > 0

) is equivalent to the compound inequality

-c \le u \le c

.

4

Subtract 1 from all parts of the compound inequality.

-5 \le -2y \le 3

This is the next step to isolate the variable

y

in the middle.

5

Divide all parts of the compound inequality by

-2

and reverse the inequality signs.

2.5 \ge y \ge -1.5

, which is equivalent to

-1.5 \le y \le 2.5

Dividing by the negative coefficient

-2

requires reversing the direction of all inequality signs.

Key Concept

Solving absolute value inequalities involving negative coefficients by isolating the absolute value and reversing inequality signs when multiplying or dividing by negative numbers.

Estimated Time:1m 30s

Question 1973Question →

For what positive integer value of $x$ is the rational expression $\frac{2x - 5}{3x^2 - 10x - 8}$ undefined?

Fill in the blanks below

The expression is undefined when

x

is equal to .

Show answer & explanation

Answer

The expression is undefined when the positive integer x is equal to 4.

The rational expression is undefined when its denominator is 0. Setting the denominator

3x^2 - 10x - 8

equal to 0 and factoring gives

(3x + 2)(x - 4) = 0

. The roots of this equation are

x = -\frac{2}{3}

and

x = 4

. Since the question restricts the target value to a positive integer, the correct value is 4.

Step-by-Step Solution

1

Set the denominator of the rational expression equal to 0.

3x^2 - 10x - 8 = 0

A rational expression is undefined when its denominator is equal to 0 because division by zero is undefined.

2

Factor the quadratic equation

3x^2 - 10x - 8 = 0

.

(3x + 2)(x - 4) = 0

Factoring the quadratic trinomial helps identify the values of x that satisfy the equation.

3

Solve for the roots by setting each factor to 0.

x = -\frac{2}{3}

or

x = 4

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.

4

Filter the solutions based on the given constraint.

x = 4

The question specifies that x must be a positive integer. Since

-\frac{2}{3}

is a negative fraction, we reject it and select 4.

Key Concept

Finding values that make a rational expression undefined by setting the denominator to zero and solving the resulting quadratic equation.

Estimated Time:1m 30s

Question 1974Question →

The matrices $A$ and $B$ are defined as follows:

A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}

What is the product matrix $AB$ ?

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}

\begin{bmatrix} 7 & 3 \\ 2 & 6 \end{bmatrix}

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

\begin{bmatrix} 6 & 0 \\ 6 & 6 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}$

Answer

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

The correct matrix is obtained by performing row-by-column matrix multiplication, which yields row 1 entries of 6 and 2, and row 2 entries of 3 and 7.

Step-by-Step Solution

1

Multiply the first row of

A

by the first column of

B

to find the element in the first row, first column of the product matrix.

2(3) + 0(0) = 6

The row-by-column rule dictates that the entry at row

i

, column

j

of the product is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Multiply the first row of

A

by the second column of

B

to find the element in the first row, second column.

2(1) + 0(2) = 2

This determines the top-right entry of the resulting matrix.

3

Multiply the second row of

A

by the first column of

B

to find the element in the second row, first column.

1(3) + 3(0) = 3

This determines the bottom-left entry of the resulting matrix.

4

Multiply the second row of

A

by the second column of

B

to find the element in the second row, second column.

1(1) + 3(2) = 1 + 6 = 7

This determines the bottom-right entry of the resulting matrix.

Key Concept

Matrix multiplication involves computing the dot product of the rows of the first matrix with the columns of the second matrix.

Question 1975Question →

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

x = -\frac{1}{2}

and

x = -2

x = 1

and

x = \frac{1}{4}

x = \frac{1}{2}

and

x = 2

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

and

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

x = \frac{3}{4}

and

x = -\frac{3}{4}

Show answer & explanation

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

Answer

The solutions are

x = \frac{1}{2}

and

x = 2

.

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

2

yields

2x^2 - 5x + 2 = 0

. Factoring this expression gives

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

, which yields the solutions

x = \frac{1}{2}

and

x = 2

.

Step-by-Step Solution

1

Multiply the entire equation by

2

to clear the fraction.

2x^2 - 5x + 2 = 0

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

2

Identify the coefficients

a

,

b

, and

c

to use in the quadratic formula.

a = 2

,

b = -5

,

c = 2

These coefficients correspond to the standard quadratic form

ax^2 + bx + c = 0

.

3

Calculate the discriminant

b^2 - 4ac

.

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

The discriminant determines the number and type of solutions.

4

Substitute the coefficients and the discriminant into the quadratic formula:

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

.

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

This formula provides the exact solutions to any quadratic equation.

5

Evaluate the two cases for the plus/minus sign.

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

and

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

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

Key Concept

Solving quadratic equations using the quadratic formula

Question 1976Question →

If the equation $3x^2 + 5x = 2$ is solved for $x$ , what is the positive difference between the two solutions?

\frac{1}{3}

\frac{4}{3}

\frac{5}{3}

\frac{7}{3}

3

Show answer & explanation

Answer: $\frac{7}{3}$

Answer

The positive difference between the two solutions is

\frac{7}{3}

.

Rearranging the equation

3x^2 + 5x = 2

by subtracting

2

from both sides gives the standard quadratic equation

3x^2 + 5x - 2 = 0

. Factoring this expression yields

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

. Setting each factor to zero gives the solutions

x = \frac{1}{3}

and

x = -2

. The positive difference between these solutions is

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

.

Step-by-Step Solution

1

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

.

3x^2 + 5x - 2 = 0

Before factoring a quadratic equation, all terms must be moved to one side so that the equation is set to zero.

2

Factor the quadratic expression by finding two binomials that multiply to

3x^2 + 5x - 2

.

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

Since the product of

3

and

-2

is

-6

, we look for factors of

-6

that sum to

5

, which are

6

and

-1

. Splitting the middle term yields

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

.

3

Use the zero-product property to solve for

x

by setting each binomial factor to zero.

3x - 1 = 0 \Rightarrow x = \frac{1}{3}

and

x + 2 = 0 \Rightarrow x = -2

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

4

Calculate the positive difference between the two solutions.

|\frac{1}{3} - (-2)| = |\frac{1}{3} + 2| = \frac{7}{3}

The positive difference is the absolute value of the subtraction of one root from the other.

Key Concept

Solving a quadratic equation by factoring over the integers after setting the equation equal to zero.

Alternative Method

Alternatively, the quadratic formula can be used. Once the equation is rewritten as

3x^2 + 5x - 2 = 0

, substitute

a=3

,

b=5

, and

c=-2

into

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

to get

x = \frac{-5 \pm \sqrt{25 - 4(3)(-2)}}{6} = \frac{-5 \pm 7}{6}

. This simplifies to

x = \frac{1}{3}

and

x = -2

. The positive difference between these values is

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

.

Estimated Time:1m 30s

Question 1977Question →

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 1978Question →

A retail electronics store sells two models of wireless headphones: Solo, which sells for $80, and Studio, which sells for$ 150. During a holiday sale, the number of headphones sold at the Downtown and Uptown store locations is represented by the matrix $N$ below, where the first row represents the Downtown store, the second row represents the Uptown store, the first column represents Solo headphones, and the second column represents Studio headphones:

N = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix}

The prices of the headphones are represented by the matrix $P$ below:

P = \begin{bmatrix} 80 \\ 150 \end{bmatrix}

Which of the following matrices represents the total revenue, in dollars, generated from headphone sales at each of the two stores, where the first row represents the Downtown store and the second row represents the Uptown store?

\begin{bmatrix} 8,000 \\ 7,300 \end{bmatrix}

\begin{bmatrix} 5,600 \\ 9,000 \end{bmatrix}

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

\begin{bmatrix} 6,850 \\ 7,650 \end{bmatrix}

\begin{bmatrix} 3,200 & 3,750 \\ 2,400 & 5,250 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}$

Answer

The column matrix with first row 6,950 and second row 7,650

The total revenue for each store is found by calculating the product of the sales matrix

N

and the price matrix

P

. This multiplication is performed by taking the dot product of each row in

N

with the column in

P

. For the first row (Downtown store), the calculation is

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

. For the second row (Uptown store), the calculation is

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

. Combining these results yields the column matrix containing 6,950 and 7,650.

Step-by-Step Solution

1

Set up the matrix multiplication to calculate the revenue, multiplying the sales matrix

N

by the price matrix

P

.

NP = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix} \begin{bmatrix} 80 \\ 150 \end{bmatrix}

To find the total revenue for each store, the number of units sold (organized by store in rows and product type in columns) must be multiplied by the unit prices (organized by product type in rows).

2

Calculate the revenue for the Downtown store by taking the dot product of the first row of

N

and the column of

P

.

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

This combines the units sold of each headphone model at the Downtown store with their respective prices to find the store's total revenue.

3

Calculate the revenue for the Uptown store by taking the dot product of the second row of

N

and the column of

P

.

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

This combines the units sold of each headphone model at the Uptown store with their respective prices to find the store's total revenue.

4

Combine the results into the final product matrix.

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

The resulting matrix contains the total revenues for the Downtown store in the first row and the Uptown store in the second row.

Key Concept

Matrix Multiplication and Modeling

Alternative Method

Instead of setting up the entire matrix multiplication, calculate the total revenue for the first store (Downtown) individually:

40 \times 80 + 25 \times 150 = 6,950

. This single calculation is enough to eliminate the options that do not have 6,950 in the first row.

Estimated Time:2m 0s

Question 1979Question →

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 1980Question →

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

Answer

Step-by-Step Solution

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