All practice questions

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

Click a left item, then click its matching right item

Show answer & explanation

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 1742Question →

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 1743Question →

For what real value of $x$ is the rational expression $\frac{2x + 1}{x - 7}$ undefined? Fill in the blank with the correct number.

Fill in the blanks below

The rational expression

\frac{2x + 1}{x - 7}

is undefined when $x = .

Show answer & explanation

Answer

The rational expression is undefined when the variable x equals 7.

A rational expression is undefined when its denominator is equal to

0

. For the expression

\frac{2x + 1}{x - 7}

, setting the denominator

x - 7 = 0

and solving for

x

yields

x = 7

.

Step-by-Step Solution

1

Set the denominator of the rational expression equal to 0.

x - 7 = 0

A rational expression is undefined when its denominator is equal to 0 because division by zero is undefined in the real number system.

2

Solve the linear equation for

x

by adding 7 to both sides.

x = 7

Isolating the variable

x

gives the value that makes the denominator zero.

Key Concept

Identifying values that make a rational expression undefined

Question 1744Question →

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 1745Question →

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 1746Question →

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 1747Question →

For what real values of $x$ is the rational expression $g(x) = \frac{x^3 - 3x^2}{\frac{2}{x-2} - \frac{x+4}{x^2 - 4}}$ undefined? Determine these three values and enter them in the blanks below in order from least to greatest.

Fill in the blanks below

The values of

x

for which

g(x)

is undefined, ordered from least to greatest, are

x =

,

x =

, and

x =

.

Show answer & explanation

Answer

The values of

x

for which the expression is undefined, ordered from least to greatest, are

-2

,

0

, and

2

.

The rational expression is undefined where any constituent denominator is equal to zero, or where the entire main denominator is equal to zero. The constituent denominators are

x-2

and

x^2-4

, which are zero when

x = 2

or

x = -2

. The entire main denominator simplifies to

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

, which is zero when

x = 0

. Therefore, the three values of

x

that make the expression undefined are

-2

,

0

, and

2

.

Step-by-Step Solution

1

Identify the values of

x

that make the denominators of the individual rational terms equal to zero.

The denominators are

x - 2

and

x^2 - 4

. Setting these to zero gives

x = 2

and

x = \pm 2

. Thus,

x = 2

and

x = -2

are values that make the individual terms undefined.

Any rational term is undefined if its denominator is equal to zero.

2

Find the values of

x

that make the overall denominator of the main fraction equal to zero by first simplifying it.

The main denominator is

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

. Finding a common denominator yields

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

. Setting this equal to zero gives

x = 0

.

The main rational expression is undefined if its entire denominator is equal to zero.

3

Combine all restricted values and list them in order from least to greatest.

The restricted values are

x = -2

,

x = 0

, and

x = 2

.

These are all the distinct real numbers that cause any division by zero in the original expression.

Key Concept

Finding the domain of nested rational expressions

Estimated Time:2m 0s

Question 1748Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq -1$ , which of the following expressions is equivalent to $\frac{3}{x} + \frac{2}{x + 1}$ ?

\frac{5}{2x + 1}

\frac{5x + 1}{x(x + 1)}

\frac{5x + 3}{x(x + 1)}

\frac{5x + 3}{2x + 1}

\frac{6}{x(x + 1)}

Show answer & explanation

Answer: $\frac{5x + 3}{x(x + 1)}$

Answer

\frac{5x + 3}{x(x + 1)}

To add the rational expressions, find the common denominator

x(x + 1)

. Multiplying the first term by

\frac{x + 1}{x + 1}

and the second term by

\frac{x}{x}

gives

\frac{3(x + 1) + 2x}{x(x + 1)}

. Expanding the numerator yields

3x + 3 + 2x

, which simplifies to

5x + 3

. Thus, the equivalent expression is

\frac{5x + 3}{x(x + 1)}

.

Step-by-Step Solution

1

Identify a common denominator for the two rational expressions.

The common denominator for the denominators

x

and

x + 1

is

x(x + 1)

.

To add fractions with different denominators, they must be written with a common denominator.

2

Rewrite each fraction with the common denominator by multiplying the numerator and denominator of the first term by

(x + 1)

and the second term by

x

.

\frac{3(x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)}

This scales the fractions to have matching denominators without changing their value.

3

Add the numerators together over the common denominator and simplify.

\frac{3x + 3 + 2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}

Combining like terms simplifies the numerator to its final form.

Key Concept

Adding rational expressions by finding a common denominator

Estimated Time:1m 0s

Question 1749Question →

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 1750Question →

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 1751Question →

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 1752Question →

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 1753Question →

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 1754Question →

Solve the equation $\sqrt{2x + 7} - x = -4$ for $x$ . What is the value of the real solution?

Show answer & explanation

Answer: 9

Answer

The only real solution to the equation is 9.

Isolating the radical yields

\sqrt{2x + 7} = x - 4

. Squaring both sides results in the quadratic equation

2x + 7 = x^2 - 8x + 16

, which simplifies to

x^2 - 10x + 9 = 0

. Factoring gives

(x - 9)(x - 1) = 0

, yielding potential solutions of 9 and 1. Checking these solutions in the original equation reveals that 9 is valid, while 1 is extraneous. Therefore, the correct real solution is 9.

Step-by-Step Solution

1

Isolate the radical on one side of the equation.

\sqrt{2x + 7} = x - 4

Before squaring both sides, the radical term must be isolated to simplify the algebraic manipulation.

2

Square both sides of the equation.

2x + 7 = (x - 4)^2

Squaring both sides eliminates the square root.

3

Expand the squared binomial.

2x + 7 = x^2 - 8x + 16

Applying the identity

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

is necessary to write the equation in polynomial form.

4

Set the quadratic equation to zero.

x^2 - 10x + 9 = 0

Subtracting

2x

and

7

from both sides allows us to solve the quadratic equation.

5

Factor the quadratic equation.

(x - 9)(x - 1) = 0

, giving potential solutions

x = 9

or

x = 1

.

Finding the roots of the quadratic equation provides the candidate solutions.

6

Verify candidates in the original equation.

The solution

x = 9

is valid, while

x = 1

is extraneous.

Squaring both sides can introduce extraneous solutions, so each candidate must be checked in the original equation.

Key Concept

Solving radical equations and verifying for extraneous solutions

Question 1755Question →

A local community center surveyed a group of $20$ students about the number of hours they volunteered last month. The table below displays the results of the survey, where $a$ and $b$ represent the number of students in their respective categories.

Hours Volunteered	Number of Students
$2$	$a$
$5$	$b$
$8$	$4$
$12$	$3$
$15$	$2$

If the mean number of hours volunteered per student for this group is $6.6$ , what is the median number of hours volunteered per student for these $20$ students?

2.0

3.5

5.0

6.6

8.0

Show answer & explanation

Answer: 5.0

Answer

The median number of hours volunteered per student is 5.0.

The correct answer is 5.0. To find this, we first establish two equations from the given information: the total count of students (

a + b + 9 = 20 \implies a + b = 11

) and the mean of the data (

\frac{2a + 5b + 32 + 36 + 30}{20} = 6.6 \implies 2a + 5b = 34

). Solving this system of equations gives

a = 7

and

b = 4

. To find the median of the 20 values, we look at the 10th and 11th sorted values. Since there are seven 2s (positions 1–7) and four 5s (positions 8–11), both the 10th and 11th values are 5, making the median 5.0.

Step-by-Step Solution

1

Set up an equation for the total number of students.

a + b = 11

The total number of students in the survey is 20. Summing the frequencies gives

a + b + 4 + 3 + 2 = 20

, which simplifies to

a + b = 11

.

2

Set up an equation for the mean of the dataset.

2a + 5b = 34

The mean of the data is 6.6. Using the formula for the weighted mean:

\frac{2a + 5b + 8(4) + 12(3) + 15(2)}{20} = 6.6

. Multiplying both sides by 20 gives

2a + 5b + 98 = 132

, which simplifies to

2a + 5b = 34

.

3

Solve the system of linear equations for

a

and

b

.

a = 7

and

b = 4

Multiply the first equation by 2 to get

2a + 2b = 22

. Subtract this from

2a + 5b = 34

to find

3b = 12 \implies b = 4

. Substitute

b = 4

back into the first equation to find

a = 7

.

4

Find the median of the 20 sorted values.

5.0

With 20 data points, the median is the average of the 10th and 11th data values in ascending order. Since there are seven 2s followed by four 5s, the 10th and 11th values are both 5. The average of 5 and 5 is 5.0.

Key Concept

Determining the median of a grouped frequency distribution by solving for missing frequencies using the total count and the mean.

Question 1756Question →

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 1757Question →

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 1758Question →

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 1759Question →

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 1760Question →

For the imaginary unit $i$ , if the complex number $z$ is defined by $z = \frac{(1 + 2i)^2}{2 - i}$ , what is the real part of $z$ ?

-2

-\frac{6}{5}

\frac{6}{5}

-\frac{10}{3}

-\frac{3}{2}

Show answer & explanation

Answer: $-2$

Answer

The real part of the complex number

z

is

-2

.

To find the real part of the complex number, we first simplify the expression by expanding the squared binomial in the numerator, which yields

-3 + 4i

. Next, we rationalize the fraction by multiplying both the numerator and the denominator by the complex conjugate of the denominator,

2 + i

. This multiplication yields

\frac{-10 + 5i}{5}

. Dividing both the real and imaginary terms by

5

results in the standard form

-2 + i

. Thus, the real part of this complex number is

-2

.

Step-by-Step Solution

1

Expand the squared binomial in the numerator of the expression for

z

.

(1 + 2i)^2 = 1^2 + 2(1)(2i) + (2i)^2 = 1 + 4i + 4i^2

Apply the algebraic identity

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

.

2

Simplify the expanded numerator using the definition of the imaginary unit.

1 + 4i + 4(-1) = -3 + 4i

Since

i^2 = -1

, the term

4i^2

simplifies to

-4

. Combining this with

1

gives the real part

-3

.

3

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator to rationalize it.

z = \frac{-3 + 4i}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{(-3 + 4i)(2 + i)}{(2 - i)(2 + i)}

Multiplying the denominator by its complex conjugate,

2 + i

, eliminates the imaginary unit from the denominator.

4

Expand and simplify the numerator and denominator.

z = \frac{-6 - 3i + 8i + 4i^2}{4 - i^2} = \frac{-6 + 5i - 4}{4 - (-1)} = \frac{-10 + 5i}{5}

Using the distributive property in the numerator gives

-6 + 5i + 4i^2

. Since

i^2 = -1

, this simplifies to

-10 + 5i

. In the denominator,

(2-i)(2+i) = 4 - i^2 = 5

.

5

Divide both terms of the simplified numerator by the denominator to express

z

in standard form

a + bi

.

z = -2 + i

Dividing the real part

-10

by

5

yields the real part

-2

, and dividing the imaginary part

5i

by

5

yields the imaginary part

i

.

Key Concept

Division of complex numbers using the complex conjugate

Estimated Time:2m 0s

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