Elementary Algebra Practice Questions — ACT — Page 2

Question 21Question →

Simplify each of the algebraic expressions on the left by distributing and combining like terms, then match it with its equivalent simplified expression on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expressions match as follows: the first expression matches

-2x^3 + 6x^2y + 4y^3

; the second expression matches

x^3 - 6x^2y + 6xy^2 + 8y^3

; the third expression matches

0

; and the fourth expression matches

y^2

.

Each expression on the left reduces to its matching counterpart on the right by carefully expanding terms (including cubes, squares, difference of squares, and difference of cubes) and combining like terms while correctly distributing negative signs.

Step-by-Step Solution

1

Simplify the first expression by expanding each term individually.

E_1 = -x^3 + 6x^2y - 12xy^2 + 8y^3 - x^3 + 4xy^2 + 8xy^2 - 4y^3

Expanding the binomial cube, the difference of squares product, and distributing the monomial allows us to identify like terms.

2

Combine like terms in the first expression.

E_1 = -2x^3 + 6x^2y + 4y^3

Combining the

x^3

,

x^2y

,

xy^2

, and

y^3

terms simplifies the expression. The

xy^2

terms sum to zero.

3

Simplify the second expression by expanding each term individually.

E_2 = 2x^3 - 4x^2y + 2xy^2 - x^3 + 8y^3 - 2x^2y + 4xy^2

Using algebraic expansion rules (including the difference of cubes product) exposes all individual terms.

4

Combine like terms in the second expression.

E_2 = x^3 - 6x^2y + 6xy^2 + 8y^3

Adding coefficients of like variable terms gives the simplified form.

5

Simplify the third expression by expanding the cubic terms.

E_3 = (x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 - 3x^2y + 3xy^2 - y^3) - (6x^2y + 2y^3) = 6x^2y + 2y^3 - 6x^2y - 2y^3

Expanding the binomial cubes and distributing the negative signs shows that all terms cancel.

6

Combine like terms in the third expression.

E_3 = 0

All terms cancel out, leaving a final value of zero.

7

Simplify the fourth expression by expanding.

E_4 = x^4 - 2x^2y + y^2 - x^4 + y^2 - y^2 + 2x^2y

Squaring the binomial, using the difference of squares, and distributing the negative variable simplifies the individual components.

8

Combine like terms in the fourth expression.

E_4 = y^2

The

x^4

and

x^2y

terms cancel, and the

y^2

terms simplify to

y^2

.

Key Concept

Simplifying Expressions and Combining Like Terms

Question 22Question →

When the expression $-\frac{1}{3}x(x - 2y)^2 - \frac{1}{2}x(xy - 3y^2) - (x^3 - 2x^2y)$ is fully simplified, what is the coefficient of the $x^2y$ term?

\frac{10}{3}

\frac{5}{6}

-\frac{7}{6}

\frac{17}{6}

\frac{3}{2}

Show answer & explanation

Answer: $\frac{17}{6}$

Answer

The coefficient of the

x^2y

term is

\frac{17}{6}

.

By expanding all components of the expression: the first term yields

-\frac{1}{3}x^3 + \frac{4}{3}x^2y - \frac{4}{3}xy^2

, the second term yields

-\frac{1}{2}x^2y + \frac{3}{2}xy^2

, and the third term yields

-x^3 + 2x^2y

. Summing the coefficients of the

x^2y

term gives

\frac{4}{3} - \frac{1}{2} + 2 = \frac{17}{6}

.

Step-by-Step Solution

1

Expand the first term of the expression.

-\frac{1}{3}x(x - 2y)^2 = -\frac{1}{3}x(x^2 - 4xy + 4y^2) = -\frac{1}{3}x^3 + \frac{4}{3}x^2y - \frac{4}{3}xy^2

Applying binomial expansion to

(x-2y)^2

and distributing

-\frac{1}{3}x

.

2

Expand the second term of the expression.

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

Distributing the term

-\frac{1}{2}x

over the parenthetical terms.

3

Distribute the negative sign in the third term.

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

Distributing the negative sign across all terms inside the parentheses.

4

Combine the coefficients of the

x^2y

terms.

\frac{4}{3} - \frac{1}{2} + 2 = \frac{8}{6} - \frac{3}{6} + \frac{12}{6} = \frac{17}{6}

Finding a common denominator of 6 to sum the coefficients of the

x^2y

term.

Key Concept

Simplifying expressions by distributing terms, expanding binomials, and combining like terms with fractional coefficients.

Question 23Question →

Solve each quadratic equation by factoring, and match the equation to its correct solution set.

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Answer

The equation

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

matches the solution set

\{-\frac{5}{3}, \frac{1}{4}\}

; the equation

x(12x + 1) = 35

matches the solution set

\{-\frac{7}{4}, \frac{5}{3}\}

; and the equation

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

matches the solution set

\{-1, \frac{4}{3}\}

.

Each of the quadratic equations is solved by first expanding any products, collecting all terms on the left-hand side to establish the standard form

ax^2 + bx + c = 0

, dividing by any common numerical factors, factoring the resulting quadratic expression into two linear binomials, and solving each linear equation for

x

. This correctly pairs the first equation with

\{-\frac{5}{3}, \frac{1}{4}\}

, the second equation with

\{-\frac{7}{4}, \frac{5}{3}\}

, and the third equation with

\{-1, \frac{4}{3}\}

.

Step-by-Step Solution

1

Solve

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

.

12x^2 + 17x - 5 = 0 \Rightarrow (3x + 5)(4x - 1) = 0 \Rightarrow x = -\frac{5}{3}

or

x = \frac{1}{4}

.

Distribute the term on the left, rearrange the terms to set the equation to zero, find factors of

12 \times (-5) = -60

that sum to

17

(which are

20

and

-3

), factor by grouping, and apply the Zero Product Property.

2

Solve

x(12x + 1) = 35

.

12x^2 + x - 35 = 0 \Rightarrow (3x - 5)(4x + 7) = 0 \Rightarrow x = \frac{5}{3}

or

x = -\frac{7}{4}

.

Expand the left side, subtract

35

from both sides, find factors of

12 \times (-35) = -420

that sum to

1

(which are

21

and

-20

), factor by grouping, and solve for

x

.

3

Solve

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

.

6x^2 - 2x - 8 = 0 \Rightarrow 3x^2 - x - 4 = 0 \Rightarrow (3x - 4)(x + 1) = 0 \Rightarrow x = \frac{4}{3}

or

x = -1

.

Expand both sides, move all terms to the left, divide the quadratic equation by

2

to simplify, factor the trinomial, and solve for the roots.

Key Concept

Rearranging non-standard quadratic equations into the standard form

ax^2 + bx + c = 0

and solving them by factoring over the integers.

Question 24Question →

Match each quadratic equation with its correct solution set by solving the equation by factoring.

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Answer

The equation

x(x - 1) = 12

matches the solution set

\{-3, 4\}

; the equation

2x^2 + 5x = 3

matches the solution set

\{-3, \frac{1}{2}\}

; the equation

3x^2 + 8 = 10x

matches the solution set

\{\frac{4}{3}, 2\}

; and the equation

2x^2 - 24 = 8x

matches the solution set

\{-2, 6\}

.

Each quadratic equation is correctly matched to its solutions by first rewriting the equation in standard form

ax^2 + bx + c = 0

, factoring the trinomial over the integers, and then applying the zero product property to find the roots.

Step-by-Step Solution

1

Set each quadratic equation to standard form

ax^2 + bx + c = 0

by expanding terms and moving all terms to one side.

The equations become:
1)

x^2 - x - 12 = 0

2)

2x^2 + 5x - 3 = 0

3)

3x^2 - 10x + 8 = 0

4)

2x^2 - 8x - 24 = 0

Before a quadratic equation can be solved by factoring, it must be set equal to zero so that the zero product property can be applied.

2

Factor each quadratic expression completely over the integers.

The factored expressions are:
1)

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

2)

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

3)

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

4)

2(x - 6)(x + 2) = 0

Factoring rewrites the quadratic expressions as products of linear factors.

3

Apply the zero product property by setting each linear factor equal to zero and solving for

x

.

The solution sets are:
1)

x = 4

or

x = -3

, yielding

\{-3, 4\}

2)

x = \frac{1}{2}

or

x = -3

, yielding

\{-3, \frac{1}{2}\}

3)

x = \frac{4}{3}

or

x = 2

, yielding

\{\frac{4}{3}, 2\}

4)

x = 6

or

x = -2

, yielding

\{-2, 6\}

If the product of two or more algebraic factors is zero, then at least one of the individual factors must equal zero.

Key Concept

Solving Quadratic Equations by Factoring

Alternative Method

You can verify the solution sets by substituting the values of the roots back into the original equations to check if they yield a true statement, or by using the quadratic formula

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

as an alternative algebraic method.

Estimated Time:1m 30s

Question 25Question →

For all real numbers $x$ and $y$ , match each algebraic expression on the left with its simplified equivalent expression on the right.

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Show answer & explanation

Answer

The expression

2(x - 3y) + 4y

simplifies to

2x - 2y

, the expression

-2(x - 3y) + 8y

simplifies to

-2x + 14y

, and the expression

2(x + 3y) - 4y

simplifies to

2x + 2y

.

Each expression on the left-hand side is expanded by applying the distributive property and then simplified by combining the terms involving

y

. This correctly matches

2(x - 3y) + 4y

to

2x - 2y

,

-2(x - 3y) + 8y

to

-2x + 14y

, and

2(x + 3y) - 4y

to

2x + 2y

.

Step-by-Step Solution

1

Simplify the expression

2(x - 3y) + 4y

.

2x - 2y

Distribute

2

to both terms inside the parentheses to get

2x - 6y

, then combine the like terms

-6y

and

4y

to get

-2y

.

2

Simplify the expression

-2(x - 3y) + 8y

.

-2x + 14y

Distribute

-2

to both terms inside the parentheses to get

-2x + 6y

, then combine the like terms

6y

and

8y

to get

14y

.

3

Simplify the expression

2(x + 3y) - 4y

.

2x + 2y

Distribute

2

to both terms inside the parentheses to get

2x + 6y

, then combine the like terms

6y

and

-4y

to get

2y

.

Key Concept

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Estimated Time:1m 0s

Question 26Question →

Match each quadratic equation with its correct set of real solutions.

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Answer

The equation

x^2 - 5x + 6 = 0

matches with the solutions

x = 2

and

x = 3

. The equation

x^2 + 5x + 6 = 0

matches with the solutions

x = -3

and

x = -2

. The equation

x^2 - x - 6 = 0

matches with the solutions

x = -2

and

x = 3

.

Each equation is solved by factoring the quadratic trinomial into two binomials, then applying the zero product property to find the values of

x

that make each factor zero. Specifically,

x^2 - 5x + 6 = 0

factors into

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

, yielding solutions

x = 2

and

x = 3

. The equation

x^2 + 5x + 6 = 0

factors into

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

, yielding solutions

x = -2

and

x = -3

. Finally,

x^2 - x - 6 = 0

factors into

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

, yielding solutions

x = 3

and

x = -2

.

Step-by-Step Solution

1

Factor the quadratic equation

x^2 - 5x + 6 = 0

.

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

Identify two integers that multiply to

6

and add up to

-5

. These integers are

-2

and

-3

.

2

Solve for

x

by setting each linear factor equal to zero:

x - 2 = 0

and

x - 3 = 0

.

x = 2

and

x = 3

Applying the zero product property means if the product of two numbers is zero, at least one of them must be zero.

3

Factor the quadratic equation

x^2 + 5x + 6 = 0

.

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

Identify two integers that multiply to

6

and add up to

5

. These integers are

2

and

3

.

4

Solve for

x

by setting each linear factor equal to zero:

x + 2 = 0

and

x + 3 = 0

.

x = -2

and

x = -3

Applying the zero product property gives the solutions as the negations of the terms inside the binomials.

5

Factor the quadratic equation

x^2 - x - 6 = 0

.

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

Identify two integers that multiply to

-6

and add up to

-1

. These integers are

-3

and

2

.

6

Solve for

x

by setting each linear factor equal to zero:

x - 3 = 0

and

x + 2 = 0

.

x = 3

and

x = -2

Setting the linear factors to zero yields the roots of the equation.

Key Concept

Solving Quadratic Equations by Factoring

Estimated Time:1m 30s

Question 27Question →

For all real values of $a$ and $b$ , what is the simplified form of the expression $3a^3b - 2ab^2 - (a^3b - 5ab^2)$ ?

2a^3b - 7ab^2

4a^3b + 3ab^2

2a^6b^2 + 3a^2b^4

2a^3b + 3ab^2

4a^3b - 7ab^2

Show answer & explanation

Answer: $2a^3b + 3ab^2$

Answer

2a^3b + 3ab^2

The correct expression is found by distributing the negative sign across the parentheses to change the signs of the terms inside, yielding

-a^3b + 5ab^2

. Combining the like terms

3a^3b

and

-a^3b

gives

2a^3b

, and combining

-2ab^2

and

+5ab^2

gives

3ab^2

.

Step-by-Step Solution

1

Distribute the negative sign to each term inside the parenthetical expression:

-(a^3b - 5ab^2)

.

-a^3b + 5ab^2

To remove the parentheses, we multiply each term inside by

-1

.

2

Rewrite the full expression with the parentheses removed:

3a^3b - 2ab^2 - a^3b + 5ab^2

.

3a^3b - 2ab^2 - a^3b + 5ab^2

This sets up the expression for combining like terms.

3

Group and combine the like terms:

(3a^3b - a^3b)

and

(-2ab^2 + 5ab^2)

.

2a^3b + 3ab^2

Only terms with the exact same variable parts and exponents can be combined by adding or subtracting their coefficients.

Key Concept

Simplifying algebraic expressions by distributing negative signs and combining like terms.

Estimated Time:45s

Question 28Question →

Match each algebraic expression on the left with its equivalent simplified form on the right. Assume all variables represent real numbers.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expression

2(3a^2 - 4b) - 3(a^2 - 2b)

matches

3a^2 - 2b

;

a(3a - b) - 2b(a - b)

matches

3a^2 - 3ab + 2b^2

;

(a + b)(3a - 2b) - b^2

matches

3a^2 + ab - 3b^2

; and

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

matches

3a^2 - ab

.

Each expression is correctly simplified by distributing coefficients, expanding binomial products, and collecting like terms.

Step-by-Step Solution

1

Simplify the expression

2(3a^2 - 4b) - 3(a^2 - 2b)

3a^2 - 2b

Distribute the coefficients to remove the parentheses:

6a^2 - 8b - 3a^2 + 6b

. Group the

a^2

terms and the

b

terms, and then combine:

(6 - 3)a^2 + (-8 + 6)b = 3a^2 - 2b

.

2

Simplify the expression

a(3a - b) - 2b(a - b)

3a^2 - 3ab + 2b^2

Distribute the variables

a

and

-2b

:

3a^2 - ab - 2ab + 2b^2

. Combine the like terms

-ab

and

-2ab

to get

-3ab

, resulting in

3a^2 - 3ab + 2b^2

.

3

Simplify the expression

(a + b)(3a - 2b) - b^2

3a^2 + ab - 3b^2

Multiply the binomial factors using the distributive property:

(a + b)(3a - 2b) = 3a^2 - 2ab + 3ab - 2b^2 = 3a^2 + ab - 2b^2

. Subtract the remaining

b^2

term:

3a^2 + ab - 2b^2 - b^2 = 3a^2 + ab - 3b^2

.

4

Simplify the expression

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

3a^2 - ab

Expand

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

. Subtract this product from

4a^2

by distributing the negative sign:

4a^2 - a^2 - ab + 2b^2

. Finally, subtract the last term

2b^2

:

3a^2 - ab + 2b^2 - 2b^2 = 3a^2 - ab

.

Key Concept

Simplifying algebraic expressions by distributing factors and combining like terms

Estimated Time:2m 0s

Question 29Question →

The polynomial $x^3 + 5x^2 - 9x - 45$ can be factored completely into three linear factors of the form $(x - a)(x - b)(x - c)$ , where $a$ , $b$ , and $c$ are integers such that $a < b < c$ . What is the value of $a - b + c$ ?

Show answer & explanation

Answer: 1

Answer

The correct answer is 1.

Factoring the polynomial by grouping gives

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

, which simplifies to

(x - 3)(x + 3)(x + 5)

after factoring the difference of squares. Writing this expression in the form

(x - a)(x - b)(x - c)

identifies the values as

3

,

-3

, and

-5

. Ordering these values to satisfy the inequality

a < b < c

yields

a = -5

,

b = -3

, and

c = 3

. Evaluating

a - b + c

gives

-5 - (-3) + 3 = 1

.

Step-by-Step Solution

1

Group the terms of the polynomial

x^3 + 5x^2 - 9x - 45

.

(x^3 + 5x^2) - (9x + 45)

Grouping allows factoring by finding common binomial terms in a cubic polynomial.

2

Factor out the greatest common factor (GCF) from each grouped term.

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

The GCF of the first group is

x^2

and the GCF of the second group is

9

.

3

Factor out the common binomial factor

(x + 5)

.

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

Both terms share the common factor

(x + 5)

.

4

Factor the quadratic term

x^2 - 9

as a difference of squares.

(x - 3)(x + 3)(x + 5)

x^2 - 9

is a difference of squares, which factors into

(x - 3)(x + 3)

.

5

Rewrite the factors in the form

(x - a)(x - b)(x - c)

to identify the values of the constants.

(x - 3)(x - (-3))(x - (-5))

which gives the set of values

\{3, -3, -5\}

.

Matching the signs of the given form

(x - \text{constant})

is necessary to correctly identify the values of the constants.

6

Sort the values in ascending order to satisfy

a < b < c

.

a = -5

,

b = -3

, and

c = 3

The inequality constraint requires sorting the values from smallest to largest.

7

Calculate the value of the expression

a - b + c

.

-5 - (-3) + 3 = 1

Substitute the sorted values into the target expression.

Key Concept

Factoring a cubic polynomial by grouping and difference of squares, and identifying algebraic constants under inequality constraints.

Question 30Question →

When the polynomial $6x^2 - 11x - 10$ is factored into the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a$ and $c$ are positive and $a > c$ , what is the value of $ad - bc$ ?

Show answer & explanation

Answer: -19

Answer

The value of

ad - bc

is

-19

.

The correct answer is

-19

. Factoring the polynomial

6x^2 - 11x - 10

yields

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

. Under the constraints that

a

and

c

are positive and

a > c

, we must have

a = 3

,

b = 2

,

c = 2

, and

d = -5

. Evaluating the expression

ad - bc

gives

(3)(-5) - (2)(2) = -15 - 4 = -19

.

Step-by-Step Solution

1

Factor the quadratic expression

6x^2 - 11x - 10

.

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

Factoring by grouping is used to rewrite the quadratic trinomial.

2

Apply the positive coefficient constraints and

a > c

to identify the constants.

a = 3

,

b = 2

,

c = 2

,

d = -5

Since the leading coefficients must be positive and

a > c

, we assign

a = 3

from the first factor and

c = 2

from the second factor.

3

Evaluate the expression

ad - bc

.

-19

Substitute the values of

a

,

b

,

c

, and

d

to calculate the final numerical value.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

where

a > 1

Question 31Question →

When the polynomial $6x^3 - 7x^2 - 16x + 12$ is factored completely into three linear factors of the form $(ax + b)(cx + d)(ex + f)$ , where $a$ , $c$ , and $e$ are positive integers, what is the value of $a + b + c + d + e + f$ ?

Show answer & explanation

Answer: 5

Answer

The value of the sum of the coefficients is 5.

The polynomial factors completely over the integers as

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

. The sum of the six coefficients is

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

.

Step-by-Step Solution

1

Find one linear factor of the cubic polynomial using the Factor Theorem.

The root

x = 2

satisfies the equation, so

(x - 2)

is a factor.

Testing integer factors of the constant term 12 reveals that

x = 2

evaluates the polynomial to 0.

2

Perform synthetic division or polynomial long division to divide the cubic by the linear factor.

The quotient is the quadratic expression

6x^2 + 5x - 6

.

This reduces the degree of the polynomial to allow quadratic factoring techniques.

3

Factor the quadratic quotient into two linear binomials.

The quadratic factors into

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

.

Using the AC method,

6 \times (-6) = -36

, and the factors of

-36

that sum to

5

are

9

and

-4

.

4

Identify the coefficients and sum them.

The sum is

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

.

The factors are

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

, corresponding to the coefficients

a=1, b=-2, c=2, d=3, e=3, f=-2

.

Key Concept

Complete factorization of cubic polynomials with integer coefficients using the Rational Root Theorem and quadratic factoring.

Question 32Question →

For each algebraic expression on the left, match it to its completely simplified equivalent expression on the right by distributing terms and combining like terms.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expressions match as follows: the first simplifies to

-3x^2y + 3y^3

; the second simplifies to

12x^2y + 6xy^2 - 2y^3

; the third simplifies to

13xy^2 - 4y^3

; and the fourth simplifies to

x^3 - 3x^2y - 6xy^2 + 2y^3

.

Each expression is expanded fully by distributing multiplication and powers, then simplified by combining terms that have the exact same variable bases and exponents.

Step-by-Step Solution

1

Simplify the first expression by distributing coefficients and combining like terms.

-3x^2y + 3y^3

Expanding the expression gives

2x^3 - 6x^2y - 3x^2y + 3y^3 - 2x^3 + 6x^2y

. Grouping the like terms:

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

, which simplifies to

-3x^2y + 3y^3

.

2

Simplify the second expression using the binomial cube formula

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

.

12x^2y + 6xy^2 - 2y^3

Expanding

(2x - y)^3

yields

8x^3 - 12x^2y + 6xy^2 - y^3

. Subtracting the remaining terms gives

8x^3 - 12x^2y + 6xy^2 - y^3 - 8x^3 + 24x^2y - y^3

. Combining like terms yields

12x^2y + 6xy^2 - 2y^3

.

3

Simplify the third expression by squaring the binomials and distributing.

13xy^2 - 4y^3

First expand the squares:

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

and

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

. Distributing the variables yields

4x^3 - 12x^2y + 9xy^2 - x^2y + 4xy^2 - 4y^3 - 4x^3 + 13x^2y

. Combining like terms results in

13xy^2 - 4y^3

.

4

Simplify the fourth expression by expanding the cubic term and distributing signs.

x^3 - 3x^2y - 6xy^2 + 2y^3

Expand

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

. Distribute all negative signs to get

2x^3 - 6x^2y - x^3 + 3x^2y - 3xy^2 + y^3 - 3xy^2 + y^3

. Combining like terms results in

x^3 - 3x^2y - 6xy^2 + 2y^3

.

Key Concept

Simplifying Expressions and Combining Like Terms

Question 33Question →

What is the sum of the solutions to the quadratic equation $x(x - 6) = 16$ ?

-6

10

6

12

38

Show answer & explanation

Answer: 6

Answer

The sum of the solutions is 6.

Rearranging the equation to standard form gives

x^2 - 6x - 16 = 0

. Factoring the trinomial yields

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

. Solving for

x

by setting each factor to zero gives

x = 8

and

x = -2

. The sum of these solutions is

8 + (-2) = 6

.

Step-by-Step Solution

1

Distribute the variable on the left side of the equation.

x^2 - 6x = 16

To solve a quadratic equation, we must first expand all products to identify the quadratic terms.

2

Subtract 16 from both sides of the equation to write it in standard form.

x^2 - 6x - 16 = 0

A quadratic equation must be set to zero before factoring.

3

Factor the quadratic trinomial.

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

We need to find two numbers that multiply to

-16

and add to

-6

. These numbers are

-8

and

2

.

4

Set each factor to zero and solve for

x

.

x = 8

and

x = -2

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

5

Add the two solutions to find their sum.

8 + (-2) = 6

The question asks for the sum of the solutions.

Key Concept

Solving quadratic equations by factoring after rearranging terms into standard form.

Question 34Question →

For all real values of $x$ , the expression $9x^2 - 36$ is equivalent to which of the following?

(3x - 6)^2

36(x - 1)(x + 1)

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

9(x - 2)(x + 2)

9(x - 2)^2

Show answer & explanation

Answer: $9(x - 2)(x + 2)$

Answer

The equivalent expression is

9(x - 2)(x + 2)

Factoring out the greatest common factor of

9

from the expression

9x^2 - 36

gives

9(x^2 - 4)

. The binomial

x^2 - 4

is a difference of squares (

x^2 - 2^2

), which can be factored into

(x - 2)(x + 2)

. Combining these parts results in the equivalent expression

9(x - 2)(x + 2)

.

Step-by-Step Solution

1

Identify the greatest common factor (GCF) of the terms in the expression

9x^2 - 36

.

The GCF of

9x^2

and

36

is

9

.

Factoring out the GCF simplifies the remaining polynomial expression.

2

Factor out the GCF of

9

from the original expression.

9(x^2 - 4)

This separates the common numeric factor from the quadratic binomial.

3

Factor the remaining binomial expression

x^2 - 4

inside the parentheses.

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

The expression

x^2 - 4

is a difference of squares (

x^2 - 2^2

), which follows the factoring pattern

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

.

4

Combine the factored components to write the final equivalent expression.

9(x - 2)(x + 2)

Combining the GCF with the factored binomial factors yields the completely factored equivalent expression.

Key Concept

Factoring a polynomial by first removing a greatest common factor and then applying the difference of squares formula.

Question 35Question →

What is the sum of the distinct real solutions to the equation $(2x - 1)^2 = (x + 2)^2$ ?

\frac{8}{3}

0

\frac{4}{3}

8

3

Show answer & explanation

Answer: $\frac{8}{3}$

Answer

The sum of the distinct real solutions is

\frac{8}{3}

.

To solve

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

, we rearrange the equation to

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

. Using the difference of squares identity, we factor this into

[(2x - 1) - (x + 2)][(2x - 1) + (x + 2)] = 0

, which simplifies to

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

. The solutions are

x = 3

and

x = -\frac{1}{3}

. Summing these gives

3 + (-\frac{1}{3}) = \frac{8}{3}

.

Step-by-Step Solution

1

Rearrange the equation by moving all terms to one side to set it to zero.

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

To solve a quadratic equation by factoring, it must first be set equal to zero.

2

Factor the expression using the difference of squares formula,

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

, where

a = 2x - 1

and

b = x + 2

.

[(2x - 1) - (x + 2)][(2x - 1) + (x + 2)] = 0

Using the difference of squares allows us to factor the quadratic expression directly without fully expanding it.

3

Simplify the terms inside each set of brackets.

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

Simplifying the binomials reveals the two linear factors of the quadratic equation.

4

Set each linear factor to zero to find the distinct real solutions.

x - 3 = 0 \Rightarrow x = 3

and

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

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

5

Calculate the sum of the distinct real solutions.

3 + \left(-\frac{1}{3}\right) = \frac{9}{3} - \frac{1}{3} = \frac{8}{3}

The question asks for the sum of the solutions.

Key Concept

Solving quadratic equations by factoring, specifically utilizing the difference of squares method after rearranging terms.

Question 36Question →

If $k$ is a positive constant and the expression $4x^2 + kx + 9$ can be written in the form $(ax + b)^2$ for some integers $a$ and $b$ , what is the value of $k$ ?

6

12

13

24

36

Show answer & explanation

Answer: 12

Answer

The value of

k

is 12.

A perfect square trinomial is of the form

(ax + b)^2 = a^2x^2 + 2abx + b^2

. Comparing this to

4x^2 + kx + 9

, we have

a^2 = 4

and

b^2 = 9

. Taking the positive roots,

a = 2

and

b = 3

. The coefficient of the middle term is

k = 2ab = 2(2)(3) = 12

. Since

k

is a positive constant, the correct value is 12.

Step-by-Step Solution

1

Identify the standard form of a perfect square trinomial.

A perfect square trinomial can be written as

(ax + b)^2 = a^2x^2 + 2abx + b^2

.

This allows us to equate the coefficients of the given expression

4x^2 + kx + 9

to the expanded form.

2

Solve for the values of

|a|

and

|b|

by equating the coefficients of

x^2

and the constant term.

a^2 = 4 \implies |a| = 2

and

b^2 = 9 \implies |b| = 3

.

The square of the first term's coefficient is

a^2

and the square of the last term's coefficient is

b^2

.

3

Calculate the middle term coefficient

k = 2ab

using the positive values since

k

is a positive constant.

k = 2 \times 2 \times 3 = 12

.

The middle term of

(ax + b)^2

is

2abx

, so the coefficient

k

must be

2ab

.

Key Concept

Factoring perfect square trinomials

Estimated Time:1m 0s

Question 37Question →

When the expression $3m(m - 2n)^2 - (2m - n)(m^2 - 3mn + 2n^2) - 4n(m^2 - mn)$ is fully simplified by combining like terms, what is the coefficient of the $m^2n$ term?

-23

-15

-9

-3

3

Show answer & explanation

Answer: $-9$

Answer

-9

The correct answer is

-9

. This is found by carefully expanding each term and distributing the negative signs: the first term expands to

3m^3 - 12m^2n + 12mn^2

; the second term, when subtracted, becomes

-2m^3 + 7m^2n - 7mn^2 + 2n^3

; and the third term simplifies to

-4m^2n + 4mn^2

. Summing the coefficients of the

m^2n

terms gives

-12 + 7 - 4 = -9

.

Step-by-Step Solution

1

Expand the first term of the expression,

3m(m - 2n)^2

.

First, square the binomial:

(m - 2n)^2 = m^2 - 4mn + 4n^2

. Next, distribute

3m

to get

3m^3 - 12m^2n + 12mn^2

.

To remove parentheses from the first term before combining like terms.

2

Expand the product in the second term,

(2m - n)(m^2 - 3mn + 2n^2)

.

2m(m^2 - 3mn + 2n^2) - n(m^2 - 3mn + 2n^2) = 2m^3 - 6m^2n + 4mn^2 - m^2n + 3mn^2 - 2n^3

. Combining like terms within this product yields

2m^3 - 7m^2n + 7mn^2 - 2n^3

.

To expand the binomial-trinomial product before applying the subtraction.

3

Subtract the expanded second term and expand the third term,

-4n(m^2 - mn)

.

Subtracting the second term gives

-2m^3 + 7m^2n - 7mn^2 + 2n^3

. Distributing the negative sign in the third term gives

-4m^2n + 4mn^2

.

To distribute the negative signs across the remaining parenthetical expressions.

4

Identify and combine all the

m^2n

terms to find the final coefficient.

The

m^2n

terms are

-12m^2n

(from the first term),

+7m^2n

(from the subtracted second term), and

-4m^2n

(from the third term). Combining these yields

(-12 + 7 - 4)m^2n = -9m^2n

.

To determine the final coefficient of the

m^2n

term.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:2m 0s

Question 38Question →

If the polynomial $12x^2 + 10x - 8$ is factored completely into the form $k(ax - 1)(bx + c)$ , where $k$ , $a$ , $b$ , and $c$ are positive integers, what is the value of $k + a + b + c$ ?

Show answer & explanation

Answer: 11

Answer

The value of

k + a + b + c

is

11

.

To factor the polynomial

12x^2 + 10x - 8

completely, we first factor out the greatest common factor of

2

, yielding

2(6x^2 + 5x - 4)

. Next, we factor the quadratic trinomial

6x^2 + 5x - 4

by finding two numbers that multiply to

6 \times (-4) = -24

and add to

5

. These numbers are

8

and

-3

. Splitting the linear term and factoring by grouping gives

6x^2 + 8x - 3x - 4 = 2x(3x + 4) - 1(3x + 4) = (2x - 1)(3x + 4)

. The completely factored expression is

2(2x - 1)(3x + 4)

. Comparing this with

k(ax - 1)(bx + c)

where

k, a, b, c

are positive integers, we determine that

k = 2

,

a = 2

,

b = 3

, and

c = 4

. Summing these values gives

2 + 2 + 3 + 4 = 11

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from the terms of the polynomial.

2(6x^2 + 5x - 4)

Factoring out the greatest common factor simplifies the coefficients, making the quadratic trinomial easier to factor.

2

Find two integers that multiply to

ac = 6 \times (-4) = -24

and add to

b = 5

.

The two numbers are

8

and

-3

.

These integers are needed to split the linear term in order to factor the quadratic by grouping.

3

Rewrite the middle term and factor the trinomial by grouping.

(2x - 1)(3x + 4)

Rewriting the trinomial as

6x^2 + 8x - 3x - 4

allows grouping of the first two terms

2x(3x + 4)

and the last two terms

-1(3x + 4)

to extract the common binomial factor.

4

Combine the factors and match the coefficients to the form

k(ax - 1)(bx + c)

.

k = 2

,

a = 2

,

b = 3

, and

c = 4

The completely factored expression is

2(2x - 1)(3x + 4)

. Matching this to the given template where all constants are positive integers yields

k = 2

,

a = 2

,

b = 3

, and

c = 4

.

5

Calculate the sum of the constants.

11

Adding the values gives

2 + 2 + 3 + 4 = 11

.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

after removing a greatest common factor.

Question 39Question →

A rectangular garden is surrounded by a uniform gravel path that is $1$ foot wide. The length of the garden is $3$ feet less than twice its width. If the total area of the garden and the path combined is $117$ square feet, what is the width of the garden, in feet?

Show answer & explanation

Answer: 7

Answer

The width of the garden is 7 feet.

By representing the garden's width as

w

, the length is

2w - 3

. The combined dimensions including the 1-foot uniform path on all sides are

w + 2

and

2w - 1

. Setting their product equal to the combined area of 117 square feet gives

(w+2)(2w-1) = 117

, which simplifies to the quadratic equation

2w^2 + 3w - 119 = 0

. Factoring this equation yields

(2w + 17)(w - 7) = 0

, giving the solutions

w = -8.5

and

w = 7

. Since width must be positive, the width of the garden is

7

feet.

Step-by-Step Solution

1

Define variables for the garden's dimensions and the combined dimensions including the path.

Garden width =

w

, garden length =

2w - 3

. Combined width =

w + 2

, combined length =

2w - 1

.

The path surrounds the garden uniformly, adding

1

foot of width to each of the four sides (adding

2

feet total to both overall width and overall length).

2

Write the area equation for the combined area.

(w + 2)(2w - 1) = 117

The total area of the garden and path combined is given as

117

square feet.

3

Expand and rearrange the equation into standard quadratic form

aw^2 + bw + c = 0

.

2w^2 + 3w - 119 = 0

Expanding

(w + 2)(2w - 1)

gives

2w^2 + 3w - 2

. Subtracting

117

from both sides yields the standard form.

4

Factor the quadratic equation over the integers.

(2w + 17)(w - 7) = 0

We find two numbers that multiply to

2 \times (-119) = -238

and sum to

3

. These numbers are

17

and

-14

. Rewriting the middle term and factoring by grouping yields

(2w + 17)(w - 7) = 0

.

5

Solve for

w

and select the mathematically and physically valid solution.

w = 7

(discarding the negative root

w = -8.5

)

A physical measurement like width must be positive.

Key Concept

Solving quadratic word problems by setting up a quadratic equation and solving it by factoring.

Estimated Time:2m 0s

Question 40Question →

When the expression $5(2x - 3y) - 3(x - 4y)$ is simplified, what is the coefficient of $y$ ?

Show answer & explanation

Answer: -3

Answer

The coefficient of

y

is

-3

.

Distributing

5

to

(2x - 3y)

yields

10x - 15y

. Distributing

-3

to

(x - 4y)

yields

-3x + 12y

. Combining the

y

terms gives

-15y + 12y = -3y

. Therefore, the coefficient of

y

is

-3

.

Step-by-Step Solution

1

Distribute the coefficients outside the parentheses.

10x - 15y - 3x + 12y

To eliminate the parentheses so that like terms can be combined.

2

Group and combine the terms containing

y

.

-15y + 12y = -3y

To determine the final simplified term containing

y

and identify its coefficient.

Key Concept

Distributing terms (especially negative coefficients) and combining like terms.

Estimated Time:45s

Elementary Algebra

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