All practice questions

For all real numbers

x

such that

x \neq -3

,

x \neq 0

, and

x \neq 3

, which of the following expressions is equivalent to

\frac{x^{-2} - 9^{-1}}{x^{-1} + 3^{-1}} \cdot \frac{3x^2 + 9x}{x - 3}?

-x - 3

x + 3

x - 3

\frac{x + 3}{3x}

-x + 3

Show answer & explanation

Answer: $-x - 3$

Answer

-x - 3

The numerator of the first factor is a difference of squares

(x^{-1} - 3^{-1})(x^{-1} + 3^{-1})

. Dividing by

x^{-1} + 3^{-1}

leaves

x^{-1} - 3^{-1} = \frac{1}{x} - \frac{1}{3} = \frac{3 - x}{3x}

. Factoring

3x

out of the numerator of the second expression gives

3x(x + 3)

. Multiplying these terms yields

\frac{3 - x}{3x} \cdot \frac{3x(x + 3)}{x - 3}

. Since

3 - x = -(x - 3)

, the binomials

(3 - x)

and

(x - 3)

cancel to

-1

, and

3x

cancels out completely, resulting in

-(x + 3) = -x - 3

.

Step-by-Step Solution

1

Factor the numerator of the first expression as a difference of squares.

x^{-2} - 9^{-1} = (x^{-1})^2 - (3^{-1})^2 = (x^{-1} - 3^{-1})(x^{-1} + 3^{-1})

Expressing negative exponents as squares allows cancellation with the denominator.

2

Divide by

(x^{-1} + 3^{-1})

and convert negative exponents into a single rational term.

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

Finding a common denominator simplifies the first factor.

3

Factor the numerator of the second expression

3x^2 + 9x

.

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

Factoring out the greatest common factor

3x

enables further simplification.

4

Multiply the simplified expressions and reduce.

\left(\frac{3 - x}{3x}\right) \cdot \left(\frac{3x(x + 3)}{x - 3}\right) = \frac{-(x - 3)}{3x} \cdot \frac{3x(x + 3)}{x - 3} = -(x + 3) = -x - 3

Canceling

3x

and noting that

(3 - x)/(x - 3) = -1

leaves the linear expression

-x - 3

.

Key Concept

Simplifying algebraic expressions containing negative exponents, rational fractions, and opposite-sign binomial factors

Question 2122Question →

A digital publishing company uses two high-speed printing presses, Press P and Press Q. Press P prints pages at a constant rate that is $40\%$ faster than the rate of Press Q. If Press P and Press Q work simultaneously at their respective constant rates, they can complete a printing job of $18,000$ pages in $5$ hours. How many hours would it take Press Q, working alone at its constant rate, to complete a job of $15,000$ pages?

Show answer & explanation

Answer: 10

Answer

10

Working together, Press P and Press Q complete

18,000

pages in

5

hours, which corresponds to a combined rate of

3,600

pages per hour. Because Press P's rate is

1.4

times Press Q's rate, their combined rate is

2.4

times Press Q's rate. Dividing

3,600

by

2.4

gives Press Q's individual rate of

1,500

pages per hour. Finally, dividing

15,000

pages by

1,500

pages per hour yields

10

hours.

Step-by-Step Solution

1

Relate the rate of Press P to Press Q

r_P = 1.4 r_Q

Press P is 40% faster than Press Q, so its rate is

1 + 0.40 = 1.4

times the rate of Press Q.

2

Calculate the combined rate expression

r_{\text{combined}} = 2.4 r_Q

When working together, their rates add:

r_P + r_Q = 1.4 r_Q + r_Q = 2.4 r_Q

.

3

Solve for the rate of Press Q (

r_Q

)

r_Q = 1,500

pages per hour

Using

\text{Work} = \text{Rate} \times \text{Time}

, we have

18,000 = (2.4 r_Q) \times 5 = 12 r_Q

. Dividing

18,000

by

12

yields

r_Q = 1,500

.

4

Determine the time required for Press Q to complete

15,000

pages

10

hours

Dividing the target workload by Press Q's rate gives

\frac{15,000\text{ pages}}{1,500\text{ pages/hour}} = 10

hours.

Key Concept

Combined Work Rates and Direct Proportions

Estimated Time:1m 30s

Question 2123Question →

When the algebraic expression $\frac{4x^3 - 16x}{2x^2 + 4x}$ is simplified for all $x \neq 0$ and $x \neq -2$ , it reduces to the linear polynomial $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: -2

Answer

The correct numerical answer is -2.

Factoring the numerator yields

4x(x - 2)(x + 2)

and factoring the denominator yields

2x(x + 2)

. Canceling the common factors

2x

and

(x + 2)

leaves

2(x - 2) = 2x - 4

. Comparing

2x - 4

to

ax + b

gives

a = 2

and

b = -4

. Summing these values gives

a + b = 2 + (-4) = -2

.

Step-by-Step Solution

1

Factor out the greatest common factor and apply the difference of squares formula to the numerator.

4x^3 - 16x = 4x(x^2 - 4) = 4x(x - 2)(x + 2)

Fully factoring the numerator allows identification of all linear factors.

2

Factor out the greatest common factor from the denominator.

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

Extracting

2x

reveals the common terms shared with the numerator.

3

Divide the numerator by the denominator by canceling identical non-zero factors

2x

and

(x + 2)

.

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

Simplifying rational expressions requires canceling common factors present in both numerator and denominator.

4

Compare the simplified expression

2x - 4

to the form

ax + b

to find

a

and

b

, then compute their sum.

a = 2

and

b = -4 \implies a + b = 2 + (-4) = -2

Matching corresponding terms identifies the values of the target constants.

Key Concept

Simplifying rational expressions by factoring out common terms and applying the difference of squares identity.

Question 2124Question →

If the quadratic equation $2x^2 - kx + 18 = 0$ has two positive real roots $x_1$ and $x_2$ such that $x_1 = 4x_2$ , what is the value of the constant $k$ ?

15

\frac{15}{2}

30

9

24

Show answer & explanation

Answer: $15$

Answer

15

By Vieta's formulas for

ax^2 + bx + c = 0

, the product of the roots is

x_1 x_2 = \frac{c}{a} = \frac{18}{2} = 9

. Given

x_1 = 4x_2

, substituting gives

4x_2^2 = 9

, which yields

x_2 = \frac{3}{2}

since roots are positive. Then

x_1 = 6

. The sum of the roots is

x_1 + x_2 = \frac{15}{2}

. By Vieta's formulas,

x_1 + x_2 = -\frac{b}{a} = \frac{k}{2}

. Setting

\frac{k}{2} = \frac{15}{2}

gives

k = 15

.

Step-by-Step Solution

1

Apply Vieta's formula for the product of the roots

x_1 \cdot x_2 = \frac{18}{2} = 9

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is

\frac{c}{a}

.

2

Substitute the given relationship

x_1 = 4x_2

into the product equation

(4x_2)(x_2) = 9 \implies 4x_2^2 = 9 \implies x_2^2 = \frac{9}{4} \implies x_2 = \frac{3}{2}

Since the roots are positive, we take the positive square root of

\frac{9}{4}

.

3

Calculate the larger root

x_1

x_1 = 4 \left(\frac{3}{2}\right) = 6

Using the relationship

x_1 = 4x_2

.

4

Apply Vieta's formula for the sum of the roots to solve for

k

x_1 + x_2 = 6 + \frac{3}{2} = \frac{15}{2} = \frac{k}{2} \implies k = 15

For

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a} = \frac{k}{2}

.

Key Concept

Relating roots of a quadratic equation to its coefficients using Vieta's formulas and factoring relationships.

Question 2125Question →

If $a$ and $b$ are the two real solutions to the quadratic equation $2x^2 - 11x + 12 = 0$ , such that $a > b$ , what is the value of $a - 2b$ ?

Show answer & explanation

Answer: 1

Answer

The value of

a - 2b

is 1.

Factoring

2x^2 - 11x + 12 = 0

yields

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

, giving solutions

x = 1.5

and

x = 4

. Given that

a > b

, we must set

a = 4

and

b = 1.5

. Substituting these values into

a - 2b

gives

4 - 2(1.5) = 1

.

Step-by-Step Solution

1

Factor the quadratic equation

2x^2 - 11x + 12 = 0

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

Find two linear factors whose product expands to

2x^2 - 11x + 12

.

2

Find the roots of the equation

x = \frac{3}{2} = 1.5

and

x = 4

Apply the zero product property:

2x - 3 = 0 \Rightarrow x = 1.5

and

x - 4 = 0 \Rightarrow x = 4

.

3

Assign values to

a

and

b

based on the inequality

a > b

a = 4

and

b = 1.5

Since

4 > 1.5

,

a

must be 4 and

b

must be 1.5.

4

Evaluate the targeted expression

a - 2b

4 - 2(1.5) = 1

Substitute

a = 4

and

b = 1.5

into

a - 2b

.

Key Concept

Factoring Quadratic Equations

Question 2126Question →

For all real numbers $x$ and $y$ such that $y \neq 0$ and $3x^2 + y^2 \neq 0$ , which of the following expressions is equivalent to $\frac{(x + y)^3 - (x - y)^3}{2y(3x^2 + y^2)}$ ?

1

\frac{y^2}{3x^2 + y^2}

\frac{3x^2}{3x^2 + y^2}

\frac{3x + y}{3x^2 + y^2}

\frac{1}{3}

Show answer & explanation

Answer: $1$

Answer

1

Expanding the numerator yields

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

. Since the numerator and denominator are identical non-zero expressions, the quotient simplifies to 1.

Step-by-Step Solution

1

Expand

(x + y)^3

and

(x - y)^3

using the binomial theorem.

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

and

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

Expanding the binomial expressions allows combination of like terms in the numerator.

2

Subtract

(x - y)^3

from

(x + y)^3

.

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

The terms

x^3

and

3xy^2

subtract to zero, while the remaining terms double.

3

Factor out common terms from the simplified numerator.

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

Factoring out

2y

reveals a factor identical to the denominator.

4

Divide the numerator by the denominator

2y(3x^2 + y^2)

.

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

Since

y \neq 0

and

3x^2 + y^2 \neq 0

, identical non-zero factors cancel out.

Key Concept

Binomial expansion of cubic expressions and factoring algebraic expressions.

Estimated Time:1m 30s

Question 2127Question →

If $m$ and $n$ are positive integers such that the quadratic equation $x^2 - mx + n = 0$ has two roots that are distinct prime numbers, and the discriminant of the equation is equal to $1$ , what is the value of $m + n$ ?

5

7

11

13

1

Show answer & explanation

Answer: 11

Answer

11

For the quadratic equation

x^2 - mx + n = 0

with roots

p_1

and

p_2

, the discriminant is

D = (p_1 - p_2)^2

. Given

D = 1

, the difference between the two prime roots is

|p_1 - p_2| = 1

. The only prime numbers with a difference of 1 are 2 and 3. Thus,

m = 2 + 3 = 5

and

n = 2 \times 3 = 6

, giving

m + n = 11

.

Step-by-Step Solution

1

Express the discriminant of the quadratic equation in terms of its roots.

If

p_1

and

p_2

are the roots of

x^2 - mx + n = 0

, Vieta's formulas give

p_1 + p_2 = m

and

p_1 p_2 = n

. The discriminant is

D = m^2 - 4n = (p_1 + p_2)^2 - 4p_1 p_2 = (p_1 - p_2)^2

.

Relating the discriminant directly to the difference of the roots simplifies the constraint.

2

Determine the roots using the given discriminant value.

Since

D = 1

, we have

(p_1 - p_2)^2 = 1

, which implies

|p_1 - p_2| = 1

.

Taking the square root of both sides indicates the two roots differ by 1.

3

Identify the prime numbers that satisfy this condition.

The only pair of prime numbers that differ by 1 is 2 and 3, because 2 is the only even prime and all other primes are odd.

Consecutive integers that are both prime must be 2 and 3.

4

Calculate

m

,

n

, and their sum

m + n

.

m = 2 + 3 = 5

and

n = 2 \times 3 = 6

. Therefore,

m + n = 5 + 6 = 11

.

Substitute the root values into the sum and product formulas.

Key Concept

Quadratic discriminant and root relationships (Vieta's Formulas)

Estimated Time:1m 30s

Question 2128Question →

If $x$ and $y$ are non-zero real numbers such that $x \neq y$ , $\frac{x^2 - y^2}{x - y} = 12$ , and $\frac{x^2y - xy^2}{xy} = 4$ , what is the value of $x^2 + y^2$ ?

Show answer & explanation

Answer: 80

Answer

The value of

x^2 + y^2

is 80.

Simplifying the first equation by factoring the difference of squares

\frac{(x-y)(x+y)}{x-y} = 12

gives

x + y = 12

. Simplifying the second equation by factoring out

xy

yields

\frac{xy(x-y)}{xy} = 4

, which gives

x - y = 4

. Solving this system yields

x = 8

and

y = 4

. Squaring and summing these values gives

8^2 + 4^2 = 64 + 16 = 80

. Alternatively, using the algebraic identity

\frac{(x+y)^2 + (x-y)^2}{2} = \frac{12^2 + 4^2}{2} = \frac{144 + 16}{2} = 80

directly yields the correct answer.

Step-by-Step Solution

1

Simplify the first given algebraic expression using the difference of squares identity.

x + y = 12

Factoring

x^2 - y^2

gives

(x - y)(x + y)

. Since

x \neq y

, we can cancel the non-zero common factor

(x - y)

from the numerator and denominator.

2

Simplify the second given algebraic expression by factoring out the greatest common factor.

x - y = 4

Factoring

xy

from

x^2y - xy^2

yields

xy(x - y)

. Since

x, y \neq 0

, we cancel

xy

from the numerator and denominator.

3

Solve for the individual values of

x

and

y

.

x = 8

and

y = 4

Adding

(x + y = 12)

and

(x - y = 4)

gives

2x = 16

, so

x = 8

. Subtracting the equations gives

2y = 8

, so

y = 4

.

4

Calculate the target expression

x^2 + y^2

.

80

x^2 + y^2 = 8^2 + 4^2 = 64 + 16 = 80

.

Key Concept

Factoring Algebraic Expressions (Difference of Squares and GCF Extraction)

Question 2129Question →

If the quadratic equation $3x^2 - 12x + c = 0$ has two real roots, $r_1$ and $r_2$ , such that $r_1^2 + r_2^2 = 10$ , what is the value of the constant $c$ ?

3

6

9

18

-9

Show answer & explanation

Answer: 9

Answer

The value of the constant

c

is 9.

The correct answer is 9. By Vieta's formulas, the sum of the roots is

r_1 + r_2 = -\frac{-12}{3} = 4

and the product of the roots is

r_1 r_2 = \frac{c}{3}

. Using the identity

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

, we substitute the given values to get

10 = 16 - \frac{2c}{3}

, which simplifies to

\frac{2c}{3} = 6

, giving

c = 9

.

Step-by-Step Solution

1

Apply Vieta's formulas to express the sum and product of the roots in terms of equation coefficients.

r_1 + r_2 = -\frac{-12}{3} = 4

and

r_1 r_2 = \frac{c}{3}

.

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

2

Relate the sum of squares

r_1^2 + r_2^2

to

(r_1 + r_2)

and

r_1 r_2

.

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

.

Expanding

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

and rearranging gives the identity for the sum of squares.

3

Substitute the known values into the identity and solve for

c

.

10 = 4^2 - 2\left(\frac{c}{3}\right) \implies 10 = 16 - \frac{2c}{3} \implies \frac{2c}{3} = 6 \implies c = 9

.

Substituting

r_1^2 + r_2^2 = 10

and

r_1 + r_2 = 4

isolates the single variable

c

.

Key Concept

Vieta's Formulas and Algebraic Identities for Quadratic Equations

Alternative Method

Alternatively, factor

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

directly once

r_1

and

r_2

are identified. Since

r_1 + r_2 = 4

and

r_1^2 + r_2^2 = 10

, solving the system of equations for

r_1

and

r_2

yields roots of 1 and 3. The product of these roots is

(1)(3) = 3

, so

\frac{c}{3} = 3

, which gives

c = 9

.

Estimated Time:1m 30s

Question 2130Question →

For all real numbers $x$ and $y$ such that $x \neq 2y$ and $x \neq -2y$ , which of the following expressions is equivalent to $\frac{x^3 + 2x^2y - 4xy^2 - 8y^3}{x^2 - 4y^2}$ ?

x - 2y

x + 4y

x + 2y

\frac{x + 2y}{x - 2y}

x^2 + 2y

Show answer & explanation

Answer: $x + 2y$

Answer

The simplified expression is

x + 2y

.

Grouping terms in the numerator gives

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

. Dividing this by the denominator

(x^2 - 4y^2)

cancels out the identical non-zero factor

(x^2 - 4y^2)

, leaving the linear binomial

x + 2y

.

Step-by-Step Solution

1

Group the four terms in the numerator in pairs to factor by grouping

x^3 + 2x^2y - 4xy^2 - 8y^3 = x^2(x + 2y) - 4y^2(x + 2y)

Grouping the first two terms and the last two terms allows factoring out

x^2

and

-4y^2

respectively.

2

Factor out the common binomial factor

(x + 2y)

from the numerator

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

Both terms share the common factor

(x + 2y)

.

3

Divide the factored numerator by the denominator

(x^2 - 4y^2)

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

Since

x \neq \pm 2y

,

x^2 - 4y^2 \neq 0

, allowing the common polynomial factor

(x^2 - 4y^2)

to be canceled.

Key Concept

Factoring Four-Term Polynomials by Grouping and Rational Expression Simplification

Question 2131Question →

If $x$ is a negative real number such that $2x^2 + 5x - 12 = 0$ , what is the value of $(2x - 3)^2$ ?

-121

0

25

55

121

Show answer & explanation

Answer: 121

Answer

121

Factoring

2x^2 + 5x - 12 = 0

gives

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

, which yields the solutions

x = \frac{3}{2}

and

x = -4

. Because the problem specifies that

x

is negative,

x

must equal

-4

. Substituting

-4

for

x

in the target expression yields

(2(-4) - 3)^2 = (-11)^2 = 121

.

Step-by-Step Solution

1

Factor the given quadratic equation to find its roots.

2x^2 + 5x - 12 = (2x - 3)(x + 4) = 0

, giving solutions

x = \frac{3}{2}

and

x = -4

.

Factoring isolates the linear terms to solve for the values of

x

.

2

Apply the given constraint to select the appropriate root.

Since

x

is specified as a negative real number,

x = -4

.

The root

x = \frac{3}{2}

is positive and violates the problem constraint.

3

Substitute the valid root into the target expression

(2x - 3)^2

.

(2(-4) - 3)^2 = (-8 - 3)^2 = (-11)^2 = 121

.

Evaluating the expression with

x = -4

gives the final requested numerical value.

Key Concept

Solving quadratic equations by factoring and evaluating algebraic expressions under given sign constraints.

Estimated Time:1m 30s

Answer

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Key Concept

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