All practice questions

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

Question 1762Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by $z = \frac{11 + 3i}{3 - i}$ . What is the real part of $z$ ?

Show answer & explanation

Answer: 3

Answer

The real part of the complex number is 3.

By multiplying both the numerator and the denominator of

\frac{11 + 3i}{3 - i}

by the conjugate of the denominator,

3 + i

, we obtain

\frac{(11+3i)(3+i)}{(3-i)(3+i)} = \frac{33 + 11i + 9i + 3i^2}{9 - i^2} = \frac{30 + 20i}{10} = 3 + 2i

. The real part of this complex number is the term without

i

, which is 3.

Step-by-Step Solution

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

3 + i

.

z = \frac{(11 + 3i)(3 + i)}{(3 - i)(3 + i)}

To eliminate the imaginary unit from the denominator.

2

Expand and simplify the numerator using the distributive property and substituting

-1

for

i^2

.

(11 + 3i)(3 + i) = 33 + 11i + 9i + 3i^2 = 33 + 20i + 3(-1) = 30 + 20i

To combine the real and imaginary terms of the numerator.

3

Expand and simplify the denominator using the difference of squares property and substituting

-1

for

i^2

.

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

To find the real number denominator.

4

Divide each term in the simplified numerator by the denominator.

z = \frac{30 + 20i}{10} = 3 + 2i

To express the complex number in the standard form

a + bi

.

5

Extract the real part of the resulting complex number

3 + 2i

.

3

The real part of a complex number in the form

a + bi

is

a

.

Key Concept

Division of complex numbers using the complex conjugate

Alternative Method

Instead of simplifying the fraction directly, assume the resulting complex number is

x + yi

, where

x

represents the real part and

y

represents the imaginary part. We can set up the equation

x + yi = \frac{11 + 3i}{3 - i}

and multiply both sides by

3 - i

to get

(x + yi)(3 - i) = 11 + 3i

. Expanding the left side gives

(3x + y) + (3y - x)i = 11 + 3i

. Equating the real and imaginary parts yields a system of linear equations:

3x + y = 11

and

-x + 3y = 3

. Multiplying the second equation by 3 and adding it to the first equation gives

10y = 20

, which means

y = 2

. Substituting

y = 2

back into the first equation yields

3x + 2 = 11

, which simplifies to

3x = 9

, or

x = 3

. The real part is therefore 3.

Estimated Time:1m 30s

Question 1763Question →

For all real numbers $x$ such that $x \neq -2$ and $x \neq 3$ , the expression $\frac{2x}{x - 3} - \frac{5}{x + 2}$ is equivalent to which of the following?

\frac{2x - 5}{2x - 1}

\frac{2x^2 - x - 15}{x^2 - x - 6}

\frac{x + 15}{x^2 - x - 6}

\frac{2x^2 - x + 15}{x^2 - x - 6}

\frac{2x^2 - x + 15}{x^2 - 6}

Show answer & explanation

Answer: \frac{2x^2 - x + 15}{x^2 - x - 6}

Answer

The expression is equivalent to

\frac{2x^2 - x + 15}{x^2 - x - 6}

.

To subtract the rational expressions, we find the common denominator, which is

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

. We rewrite each fraction with this denominator, which gives

\frac{2x(x + 2)}{x^2 - x - 6} = \frac{2x^2 + 4x}{x^2 - x - 6}

and

\frac{5(x - 3)}{x^2 - x - 6} = \frac{5x - 15}{x^2 - x - 6}

. Subtracting the second numerator from the first gives

(2x^2 + 4x) - (5x - 15) = 2x^2 - x + 15

. Placing this over the common denominator gives the simplified expression

\frac{2x^2 - x + 15}{x^2 - x - 6}

.

Step-by-Step Solution

1

Identify the common denominator.

The common denominator is

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

.

To subtract rational expressions, we need a common denominator.

2

Rewrite each rational expression with the common denominator.

The first term becomes

\frac{2x(x + 2)}{(x - 3)(x + 2)} = \frac{2x^2 + 4x}{x^2 - x - 6}

, and the second term becomes

\frac{5(x - 3)}{(x - 3)(x + 2)} = \frac{5x - 15}{x^2 - x - 6}

.

Multiplying the numerator and denominator of each term by the missing factor keeps the values of the expressions unchanged.

3

Subtract the numerators.

(2x^2 + 4x) - (5x - 15) = 2x^2 + 4x - 5x + 15 = 2x^2 - x + 15

.

Subtracting the second numerator requires distributing the negative sign to both terms of the expression

(5x - 15)

.

4

Combine the result over the common denominator.

\frac{2x^2 - x + 15}{x^2 - x - 6}

Write the simplified numerator over the common denominator.

Key Concept

Subtraction of rational expressions involves finding a common denominator, expanding the numerators, and distributing negative signs carefully.

Question 1764Question →

For all real numbers $x$ and $y$ , when the expression $-2x(x - 3y)^2 - y^2(4x - y) - 3x^2(2x - y)$ is completely simplified, what is the coefficient of the $x^2y$ term?

-9

3

9

12

15

Show answer & explanation

Answer: 15

Answer

15

Completely simplifying the given expression yields

-8x^3 + 15x^2y - 22xy^2 + y^3

. The coefficient of the

x^2y

term is

15

, which is obtained by combining the term

12x^2y

(from the distribution of

-2x

over the middle term of the squared binomial) and the term

3x^2y

(from the distribution of

-3x^2

over

-y

).

Step-by-Step Solution

1

Expand the binomial squared expression

(x - 3y)^2

.

(x - 3y)^2 = x^2 - 6xy + 9y^2

Following the order of operations, we must square the binomial before distributing the outer term.

2

Distribute

-2x

to the trinomial result from the previous step.

-2x(x^2 - 6xy + 9y^2) = -2x^3 + 12x^2y - 18xy^2

Using the distributive property, we multiply coefficients and add exponents of like bases (noting that

-2x \times -6xy = 12x^2y

).

3

Distribute

-y^2

to the binomial

(4x - y)

.

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

We multiply each term inside the parentheses by

-y^2

, keeping track of the signs.

4

Distribute

-3x^2

to the binomial

(2x - y)

.

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

We distribute the

-3x^2

factor, ensuring that multiplying two negative values results in a positive term (

-3x^2 \times -y = 3x^2y

).

5

Combine the coefficients of all like terms containing

x^2y

.

12x^2y + 3x^2y = 15x^2y

We add the coefficients of the terms that share the exact variable part

x^2y

to find the final coefficient.

Key Concept

Simplifying algebraic expressions by distributing terms, applying exponent rules, and combining like terms.

Question 1765Question →

If $\sqrt{3x - 2} = 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x

is 6.

Squaring both sides of the equation

\sqrt{3x - 2} = 4

eliminates the radical, leading to

3x - 2 = 16

. Adding 2 to both sides results in

3x = 18

. Dividing both sides by 3 gives

x = 6

. Substituting 6 back into the original equation yields

\sqrt{3(6) - 2} = \sqrt{18 - 2} = \sqrt{16} = 4

, which verifies that the solution is correct.

Step-by-Step Solution

1

Square both sides of the equation to remove the radical.

3x - 2 = 16

Squaring a square root removes the radical sign since

(\sqrt{a})^2 = a

for non-negative values.

2

Add 2 to both sides of the equation.

3x = 18

Adding the constant term moves it to the other side to isolate the term containing the variable.

3

Divide both sides by 3.

x = 6

Dividing by the coefficient of

x

yields the final solution.

Key Concept

Solving basic radical equations by isolating the radical and squaring both sides.

Question 1766Question →

For all real values of $a$ and $b$ , which of the following is equivalent to the expression $2a(a^2 - 3b) - 3(a^3 - 2ab + b^2) - (4b^2 - a^3)$ ?

-2a^3 - 7b^2

-12ab - 7b^2

-b^2

-7b^2

-2a^3 + 2a^2 - 7b^2

Show answer & explanation

Answer: $-7b^2$

Answer

-7b^2

Distributing the terms yields

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

. Combining the like terms for

a^3

gives

2a^3 - 3a^3 + a^3 = 0

. Combining the

ab

terms gives

-6ab + 6ab = 0

. Combining the

b^2

terms gives

-3b^2 - 4b^2 = -7b^2

. Thus, the simplified expression is

-7b^2

.

Step-by-Step Solution

1

Distribute the factors outside the parentheses to each term inside the parentheses.

The terms expand to:

2a(a^2) - 2a(3b) - 3(a^3) - 3(-2ab) - 3(b^2) - 1(4b^2) - 1(-a^3) = 2a^3 - 6ab - 3a^3 + 6ab - 3b^2 - 4b^2 + a^3

Distribution eliminates parentheses, making it possible to group and combine like terms.

2

Group like terms together based on their variable parts and powers.

(2a^3 - 3a^3 + a^3) + (-6ab + 6ab) + (-3b^2 - 4b^2)

Grouping like terms makes it easier to perform the arithmetic on the coefficients.

3

Combine the coefficients for each group of like terms.

0a^3 + 0ab - 7b^2 = -7b^2

Simplifying the coefficient sums yields the final simplified form of the expression.

Key Concept

Simplifying expressions by distributing terms and combining like terms

Question 1767Question →

For the imaginary unit $i$ , where $i^2 = -1$ , which of the following is equivalent to the expression $\frac{5}{2 - i}$ ?

2 - i

10 + 5i

2 + i

\frac{10 + 5i}{3}

2 + 5i

Show answer & explanation

Answer: $2 + i$

Answer

The simplified expression is

2 + i

.

To simplify the expression, multiply both the numerator and denominator by the complex conjugate of the denominator, which is

2 + i

. This results in

\frac{5(2+i)}{(2-i)(2+i)} = \frac{10+5i}{4-i^2}

. Since

i^2 = -1

, the denominator becomes

4 - (-1) = 5

. Dividing both terms in the numerator by

5

gives

2 + i

.

Step-by-Step Solution

1

Multiply the numerator and the denominator by the complex conjugate of the denominator,

2 + i

.

\frac{5(2 + i)}{(2 - i)(2 + i)}

Multiplying by the conjugate rationalizes the denominator, converting it into a real number.

2

Expand the numerator and the denominator, substituting

-1

for

i^2

.

\frac{10 + 5i}{4 - (-1)} = \frac{10 + 5i}{5}

Using the distributive property for the numerator and the difference of squares identity for the denominator, along with the definition

i^2 = -1

.

3

Divide each term in the numerator by the denominator.

2 + i

Distributing the division by

5

to both the real and imaginary parts of the numerator simplifies the expression to standard form.

Key Concept

Rationalizing the denominator of a complex fraction by multiplying by the complex conjugate of the denominator.

Estimated Time:45s

Question 1768Question →

For the imaginary unit $i$ , where $i^2 = -1$ , let $z$ be the complex number defined by $z = \frac{10 + ki}{2 - i}$ , where $k$ is a real constant. If the imaginary part of $z$ is $4$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is

5

.

Multiplying the numerator and denominator of

z = \frac{10 + ki}{2 - i}

by the complex conjugate

2 + i

gives

z = \frac{(20 - k) + (10 + 2k)i}{5}

. The imaginary part is

\frac{10 + 2k}{5}

. Setting this expression equal to

4

and solving for

k

yields

k = 5

.

Step-by-Step Solution

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

2 + i

.

z = \frac{(10 + ki)(2 + i)}{(2 - i)(2 + i)}

To eliminate the imaginary unit from the denominator and express the complex number in standard form.

2

Expand both the numerator and the denominator, using the property

i^2 = -1

.

z = \frac{20 + 10i + 2ki + ki^2}{4 - i^2} = \frac{(20 - k) + (10 + 2k)i}{5}

To separate the real terms and imaginary terms in the numerator and simplify the denominator to a real number.

3

Express the complex number in standard form

a + bi

to identify the imaginary part.

z = \frac{20 - k}{5} + \left(\frac{10 + 2k}{5}\right)i

The imaginary part of a complex number is the coefficient of

i

, which is

\frac{10 + 2k}{5}

.

4

Set the imaginary part equal to

4

and solve the linear equation for

k

.

\frac{10 + 2k}{5} = 4 \implies 10 + 2k = 20 \implies 2k = 10 \implies k = 5

To find the specific value of the constant

k

that makes the imaginary part of

z

equal to

4

.

Key Concept

Rationalizing complex numbers and identifying real and imaginary components

Question 1769Question →

The quadratic equation $x^2 - 4x - 12 = 0$ has two real solutions. What is the value of the positive solution to this equation?

Show answer & explanation

Answer: 6

Answer

The positive solution to the equation is

6

.

Factoring the quadratic trinomial

x^2 - 4x - 12 = 0

yields

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

. Setting the individual binomial factors to zero gives the solutions

x = 6

and

x = -2

. The positive solution among these is

6

.

Step-by-Step Solution

1

Factor the quadratic equation

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

Factoring the trinomial

x^2 - 4x - 12

requires finding two integers whose product is

-12

and whose sum is

-4

. These numbers are

-6

and

2

.

2

Apply the zero product property

x - 6 = 0

or

x + 2 = 0

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

3

Solve for the variable and identify the positive root

x = 6

and

x = -2

Solving the linear equations yields

x = 6

and

x = -2

. Since the question asks for the positive solution, we select

6

.

Key Concept

Solving quadratic equations by factoring

Question 1770Question →

For all real values of

x

where the expression is defined, consider the equation:

\frac{2x}{\sqrt{x + 3}} = \sqrt{x + 3} - 1

Which of the following represents the complete set of real solutions to this equation?

\{1\}

\{1, 6\}

\{6\}

\{-3, 4\}

\{2\}

Show answer & explanation

Answer: $\{1\}$

Answer

The set containing only 1

The correct answer is the set containing only 1. To solve the equation, we first multiply both sides by the denominator

\sqrt{x+3}

, which yields

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

. Isolating the radical gives

\sqrt{x+3} = 3-x

. Squaring both sides produces the quadratic equation

x+3 = x^2-6x+9

, which simplifies to

x^2-7x+6=0

. Solving this quadratic gives candidate solutions of 1 and 6. Substituting 6 back into the original equation results in an invalid statement (

4 = 2

), making it extraneous. Substituting 1 yields a valid statement (

1 = 1

), meaning the only real solution is 1.

Step-by-Step Solution

1

Determine the domain of the equation.

x > -3

The expression inside the square root must be non-negative (

x+3 \ge 0

), and since it is in the denominator, it cannot be zero (

x+3 \neq 0

).

2

Clear the denominator by multiplying both sides by

\sqrt{x + 3}

.

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

Multiplying both sides by the denominator simplifies the rational expression into a form where we can isolate the radical.

3

Isolate the radical term.

\sqrt{x + 3} = 3 - x

Grouping all non-radical terms on one side prepares the equation for squaring to eliminate the radical.

4

Analyze constraints on the variable.

x \le 3

Since the principal square root on the left side is non-negative, the right side

3 - x

must also be non-negative, which restricts any valid solutions to

x \le 3

.

5

Square both sides and simplify to form a quadratic equation.

x^2 - 7x + 6 = 0

Squaring both sides eliminates the radical:

x + 3 = (3 - x)^2 \implies x + 3 = x^2 - 6x + 9 \implies x^2 - 7x + 6 = 0

.

6

Solve the quadratic equation by factoring.

x = 1

or

x = 6

Factoring

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

gives the potential solutions.

7

Verify the solutions in the original equation.

x = 1

is valid;

x = 6

is extraneous.

Substituting

x = 6

yields

\frac{12}{3} = 4

on the left and

\sqrt{9}-1 = 2

on the right, which are not equal. Substituting

x = 1

yields

\frac{2}{2} = 1

on the left and

\sqrt{4}-1 = 1

on the right, which are equal.

Key Concept

Solving equations containing both rational and radical expressions requires clearing denominators, isolating the radical, squaring both sides, and verifying candidate solutions to eliminate extraneous solutions.

Question 1771Question →

Which of the following is the complete set of real solutions to the equation $\sqrt{5x - 9} = x - 3$ ?

\{9\}

\{2, 9\}

\{0, 5\}

\{1.5, 2\}

\{2\}

Show answer & explanation

Answer: $\{9\}$

Answer

\{9\}

The correct option is the set containing only the value 9. To solve the equation, we square both sides to get

5x - 9 = x^2 - 6x + 9

, which simplifies to the quadratic equation

x^2 - 11x + 18 = 0

. Factoring this yields the potential solutions 9 and 2. Substituting these back into the original equation, we find that 9 satisfies the equation while 2 results in a contradiction, making it extraneous.

Step-by-Step Solution

1

Isolate the radical and square both sides of the equation.

5x - 9 = (x - 3)^2

Squaring both sides eliminates the square root to allow solving for the variable.

2

Expand the right-hand side using the binomial squaring rule.

5x - 9 = x^2 - 6x + 9

The square of a binomial

(a - b)^2

is

a^2 - 2ab + b^2

.

3

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

x^2 - 11x + 18 = 0 \Rightarrow (x - 9)(x - 2) = 0

Rearranging terms simplifies the equation into a standard quadratic form that can be factored.

4

Solve for the potential roots and check for extraneous solutions in the original equation.

x = 9

(valid) and

x = 2

(extraneous)

Checking

x = 9

gives

\sqrt{5(9) - 9} = 9 - 3 \Rightarrow 6 = 6

(true). Checking

x = 2

gives

\sqrt{5(2) - 9} = 2 - 3 \Rightarrow 1 = -1

(false).

Key Concept

Solving radical equations and verifying for extraneous solutions

Question 1772Question →

A rectangular prism has a volume represented by the expression $3x^3 - 5x^2 - 12x + 20$ cubic centimeters. If the height of the prism is $x - 2$ centimeters, which of the following expressions represents a possible length of the base of the prism, in centimeters, assuming the length and width are linear binomials with integer coefficients?

x - 2

3x^2 - 5

3x - 5

3x - 10

x + 4

Show answer & explanation

Answer: $3x - 5$

Answer

The correct answer is the expression

3x - 5

.

The polynomial representing the volume can be factored by grouping:

3x^3 - 5x^2 - 12x + 20 = x^2(3x - 5) - 4(3x - 5) = (3x - 5)(x^2 - 4)

. Factoring the difference of squares yields

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

. Since the height is

x - 2

, the remaining dimensions of the base must be

3x - 5

and

x + 2

. Therefore, the expression

3x - 5

is a possible length of the base.

Step-by-Step Solution

1

Group the terms of the cubic polynomial representing the volume:

3x^3 - 5x^2 - 12x + 20

.

(3x^3 - 5x^2) - (12x - 20)

Grouping terms allows us to factor the polynomial by grouping.

2

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

x^2(3x - 5) - 4(3x - 5)

The GCF of

3x^3

and

5x^2

is

x^2

, and the GCF of

12x

and

20

is

4

.

3

Factor out the common binomial factor

(3x - 5)

.

(3x - 5)(x^2 - 4)

This rewrites the polynomial as a product of a linear binomial and a quadratic binomial.

4

Factor the quadratic term

x^2 - 4

using the difference of squares identity.

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

The expression

x^2 - 4

is a difference of squares, which factors into

(x - 2)(x + 2)

.

5

Divide the factored volume by the height,

x - 2

, to find the possible dimensions of the base.

The possible dimensions for the length and width of the base are

3x - 5

and

x + 2

.

Volume is the product of length, width, and height. Since the height is

x - 2

, the remaining factors represent the length and width.

Key Concept

Factoring polynomials by grouping and difference of squares.

Estimated Time:2m 0s

Question 1773Question →

Which of the following is a factor of the expression $2x^4 - 8x^2y^2 - 2x^2 + 8y^2$ when it is factored completely?

(x - 2y)^2

x^2 - 8y^2

x - 2y

x^2 + 2y^2

x^2 - 2y

Show answer & explanation

Answer: $x - 2y$

Answer

The correct answer is

x - 2y

because the completely factored form of the expression is

2(x - 1)(x + 1)(x - 2y)(x + 2y)

, which contains

x - 2y

as a linear factor.

The expression

2x^4 - 8x^2y^2 - 2x^2 + 8y^2

can be factored by first pulling out the greatest common factor of 2, giving

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

. Grouping the terms as

x^2(x^2 - 4y^2) - 1(x^2 - 4y^2)

produces

2(x^2 - 1)(x^2 - 4y^2)

. Factoring the differences of squares yields the completely factored form

2(x - 1)(x + 1)(x - 2y)(x + 2y)

. The expression

x - 2y

is one of these linear factors.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) of the terms in the polynomial.

The terms

2x^4

,

-8x^2y^2

,

-2x^2

, and

8y^2

share a common factor of 2. Factoring out 2 yields:

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

.

Factoring out the GCF simplifies the remaining polynomial expression, making it easier to factor further.

2

Group the terms inside the parentheses to perform factoring by grouping.

Group the terms as follows:

2[(x^4 - 4x^2y^2) - (x^2 - 4y^2)]

. Factor out

x^2

from the first group:

2[x^2(x^2 - 4y^2) - 1(x^2 - 4y^2)]

. Now, factor out the common binomial

(x^2 - 4y^2)

to get:

2(x^2 - 1)(x^2 - 4y^2)

.

Grouping allows us to find common binomial factors within the terms of the polynomial.

3

Apply the difference of squares identity,

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

, to the remaining binomial factors.

For the factor

(x^2 - 1)

, the difference of squares gives

(x - 1)(x + 1)

. For the factor

(x^2 - 4y^2)

, the difference of squares gives

(x - 2y)(x + 2y)

. Substituting these back into the expression yields:

2(x - 1)(x + 1)(x - 2y)(x + 2y)

.

Both quadratic factors are differences of squares and must be factored completely to find all linear factors.

4

Compare the complete factorization with the given choices to find the matching factor.

The linear factor

x - 2y

is present in the completely factored expression.

This confirms the correct option based on algebraic factorization.

Key Concept

Factoring polynomials completely using GCF, grouping, and the difference of squares identity.

Alternative Method

Instead of factoring out the GCF 2 first, you can group the terms directly:

2x^4 - 2x^2 - 8x^2y^2 + 8y^2 = 2x^2(x^2 - 1) - 8y^2(x^2 - 1) = (2x^2 - 8y^2)(x^2 - 1)

. Then, factor out 2 from the first binomial to get

2(x^2 - 4y^2)(x^2 - 1)

, and finally apply the difference of squares identity to both quadratic factors to obtain

2(x - 2y)(x + 2y)(x - 1)(x + 1)

.

Estimated Time:1m 30s

Question 1774Question →

A certain real number $x$ satisfies the condition that the square of $3$ less than twice $x$ is equal to $8$ times the quantity $3$ minus $x$ . What is the ratio of the larger solution to the smaller solution of this equation?

-\frac{15}{4}

-\frac{5}{3}

-\frac{3}{5}

\frac{5}{3}

3

Show answer & explanation

Answer: $-\frac{5}{3}$

Answer

The ratio of the larger solution to the smaller solution is

-\frac{5}{3}

.

The correct answer is

-\frac{5}{3}

. The verbal statement translates directly to

(2x - 3)^2 = 8(3 - x)

. Expanding both sides yields

4x^2 - 12x + 9 = 24 - 8x

. Rearranging into standard form gives

4x^2 - 4x - 15 = 0

. Factoring by grouping yields

(2x + 3)(2x - 5) = 0

, giving the solutions

x = -\frac{3}{2}

and

x = \frac{5}{2}

. The ratio of the larger root to the smaller root is

\frac{5/2}{-3/2} = -\frac{5}{3}

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

(2x - 3)^2 = 8(3 - x)

'Twice

x

' is

2x

, '3 less than twice

x

' is

2x - 3

, and its square is

(2x - 3)^2

. This is equal to 8 times the quantity

3 - x

.

2

Expand both sides of the equation.

4x^2 - 12x + 9 = 24 - 8x

Expanding the binomial

(2x - 3)^2

gives

4x^2 - 12x + 9

and distributing the right side gives

24 - 8x

.

3

Rearrange the equation to set it equal to zero.

4x^2 - 4x - 15 = 0

Add

8x

and subtract

24

from both sides to gather all terms on one side of the equation.

4

Factor the quadratic equation by grouping.

(2x + 3)(2x - 5) = 0

Find two numbers that multiply to

4 \times (-15) = -60

and add to

-4

. These numbers are

-10

and

6

. Rewrite the middle term as

-10x + 6x

and factor:

2x(2x - 5) + 3(2x - 5) = 0 \implies (2x + 3)(2x - 5) = 0

.

5

Solve for the roots of the equation.

x = -\frac{3}{2}

and

x = \frac{5}{2}

Set each factor equal to zero:

2x + 3 = 0 \implies x = -\frac{3}{2}

and

2x - 5 = 0 \implies x = \frac{5}{2}

.

6

Identify the larger and smaller solutions and compute their ratio.

-\frac{5}{3}

The larger solution is

\frac{5}{2}

and the smaller solution is

-\frac{3}{2}

. Their ratio is

\frac{5/2}{-3/2} = -\frac{5}{3}

.

Key Concept

Solving quadratic equations of the form

ax^2 + bx + c = 0

by factoring over the integers.

Question 1775Question →

The trinomial $2x^2 + 7x + 3$ can be factored into the product of two binomials of the form $(2x + a)(x + b)$ , where $a$ and $b$ are integers. What is the value of the expression $a + 2b$ ?

Show answer & explanation

Answer: 7

Answer

The value of the expression

a + 2b

is 7.

Expanding the factored template

(2x + a)(x + b)

yields

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

. Comparing this to the given expression

2x^2 + 7x + 3

, the coefficient of

x

on the left side is

a + 2b

, and on the right side is 7. Therefore,

a + 2b = 7

. Alternatively, factoring

2x^2 + 7x + 3

yields

(2x + 1)(x + 3)

, where

a = 1

and

b = 3

. Substituting these integers into

a + 2b

gives

1 + 2(3) = 7

.

Step-by-Step Solution

1

Expand the expression

(2x + a)(x + b)

using the FOIL method.

2x^2 + 2bx + ax + ab = 2x^2 + (a + 2b)x + ab

Expanding the template allows direct comparison of its coefficients with the given trinomial.

2

Equate the coefficients of the expanded template to the given trinomial

2x^2 + 7x + 3

.

a + 2b = 7

and

ab = 3

For the two polynomial expressions to be equivalent for all values of

x

, their corresponding coefficients must be equal.

3

Identify the requested value directly from the system of equations.

7

The question asks for the value of

a + 2b

, which is precisely the coefficient of the linear

x

term.

Key Concept

Factoring quadratic trinomials with a leading coefficient greater than 1

Alternative Method

Factor the trinomial

2x^2 + 7x + 3

using the AC method: multiply the leading coefficient (2) and the constant term (3) to get 6. Find two numbers that multiply to 6 and add to 7, which are 6 and 1. Rewrite the middle term:

2x^2 + 6x + x + 3

. Factor by grouping:

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

. Compare this to

(2x + a)(x + b)

to find

a = 1

and

b = 3

, then compute

a + 2b = 1 + 2(3) = 7

.

Estimated Time:45s

Question 1776Question →

For all real values of $x$ and $y$ , the expression $3x(x - 2y) - 2(x^2 - 4xy + y^2) - x^2$ can be written in the form $axy + by^2$ , where $a$ and $b$ are constants. What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of

a

is 2.

The value of

a

is 2 because distributing

3x(x - 2y)

yields

3x^2 - 6xy

, and distributing

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

yields

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

. Combining these with the

-x^2

term yields

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

. Comparing this to

axy + by^2

shows that

a

, the coefficient of the

xy

term, is 2.

Step-by-Step Solution

1

Distribute

3x

across the first parenthetical expression

(x - 2y)

3x^2 - 6xy

To clear the first set of parentheses by multiplying

3x

by each term inside.

2

Distribute

-2

across the second parenthetical expression

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

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

To clear the second set of parentheses. Note that multiplying

-2

by

-4xy

yields a positive term

+8xy

due to the sign rules.

3

Write the full expression and group like terms

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

To group terms with identical variable parts so they can be combined.

4

Combine the coefficients of the grouped terms

2xy - 2y^2

Simplifying the groups:

3-2-1=0

for the

x^2

terms, and

-6+8=2

for the

xy

terms.

5

Compare the simplified expression to the form

axy + by^2

to find the coefficient

a

a = 2

The coefficient of the

xy

term is

2

, which corresponds to

a

in the target expression.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

Question 1777Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $w$ is defined as $w = \frac{6 + 4i}{2i}$ . What is the imaginary part of $w$ ?

Show answer & explanation

Answer: -3

Answer

The imaginary part of

w

is

-3

.

Dividing each term in the numerator of

\frac{6 + 4i}{2i}

by the denominator

2i

yields

\frac{6}{2i} + \frac{4i}{2i}

, which simplifies to

\frac{3}{i} + 2

. Since

i^2 = -1

, the term

\frac{3}{i}

can be rationalized to

-3i

. Thus, the complex number in standard form is

2 - 3i

. The imaginary part is the real coefficient of

i

, which is

-3

.

Step-by-Step Solution

1

Divide each term in the numerator by the denominator.

w = \frac{6}{2i} + \frac{4i}{2i}

This separates the quotient into two simpler terms that can be simplified individually.

2

Simplify both terms.

w = \frac{3}{i} + 2

Reduce the fractions by dividing out common factors in both the numerators and the denominators.

3

Rationalize the denominator of the imaginary term.

\frac{3}{i} \cdot \frac{i}{i} = \frac{3i}{i^2} = \frac{3i}{-1} = -3i

Multiply the numerator and denominator by

i

to eliminate the imaginary unit from the denominator, using the property

i^2 = -1

.

4

Combine the real and imaginary parts into standard form

a + bi

.

w = 2 - 3i

Group the real constant and the simplified imaginary term together.

5

Identify the imaginary part of the complex number.

-3

The imaginary part of a complex number

a + bi

is the real coefficient

b

of the imaginary unit

i

.

Key Concept

Simplifying a quotient of complex numbers by dividing by a pure imaginary number.

Question 1778Question →

What is the sum of all real values of $x$ that satisfy the equation $(x - 3)^2 + x(x + 2) = 15$ ?

-2

-1

0

2

4

Show answer & explanation

Answer: 2

Answer

The sum of all real values of

x

that satisfy the equation is

2

.

Expanding the equation yields

2x^2 - 4x + 9 = 15

. Setting this to zero gives

2x^2 - 4x - 6 = 0

. Dividing by the common factor of

2

simplifies this to

x^2 - 2x - 3 = 0

. Factoring the trinomial yields

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

, which gives the two solutions

x = 3

and

x = -1

. Summing these two solutions gives

3 + (-1) = 2

.

Step-by-Step Solution

1

Expand both terms on the left side of the equation.

(x^2 - 6x + 9) + (x^2 + 2x) = 15

Applying the binomial squaring formula

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

to

(x - 3)^2

and distributing

x

to both terms in

x(x + 2)

allows us to simplify the equation.

2

Combine like terms and set the quadratic equation to zero.

2x^2 - 4x - 6 = 0

Grouping

x^2

terms,

x

terms, and constant terms on one side is necessary to format the quadratic equation as

ax^2 + bx + c = 0

before factoring.

3

Divide the entire equation by the common factor of

2

to simplify factoring.

x^2 - 2x - 3 = 0

Simplifying the quadratic equation makes it easier to find two binomial factors.

4

Factor the quadratic trinomial by finding two numbers that multiply to

-3

and add to

-2

.

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

Since

-3 \times 1 = -3

and

-3 + 1 = -2

, we can write the quadratic in factored form.

5

Set each factor to zero to solve for

x

.

x = 3

or

x = -1

Applying the zero product property determines the two values of

x

that satisfy the original equation.

6

Calculate the sum of the two solutions.

3 + (-1) = 2

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Solving quadratic equations by rearranging terms, factoring trinomials, and applying the zero product property.

Question 1779Question →

For all real values of $x$ and $y$ , which of the following is equivalent to the expression $x(x - y)^2 - x^2(x - 2y)$ ?

xy^2

2x^2y + xy^2

-4x^2y + xy^2

-x^3 + x^2 + xy^2

y^2

Show answer & explanation

Answer: $xy^2$

Answer

The simplified expression is

xy^2

.

Expanding

(x - y)^2

yields

x^2 - 2xy + y^2

. Distributing

x

to this expression results in

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

. Distributing

-x^2

to

(x - 2y)

yields

-x^3 + 2x^2y

. Combining these parts gives

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

, which simplifies completely to

xy^2

.

Step-by-Step Solution

1

Expand the squared binomial

(x - y)^2

using the algebraic identity

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

.

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

Expanding the binomial is necessary before distributing the outer variable.

2

Distribute the term

x

to each term in the expanded binomial, and distribute the term

-x^2

to each term inside the second parenthesis.

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

and

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

Distribution eliminates parentheses and prepares the expression for combining like terms.

3

Combine all like terms in the resulting expression:

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

.

xy^2

Combining like terms simplifies the expression to its final equivalent form.

Key Concept

Simplifying algebraic expressions by expanding binomials, distributing variables, and combining like terms.

Estimated Time:1m 0s

Question 1780Question →

If $x$ is a real number that satisfies the equation $\sqrt{2x + 7} + \sqrt{x + 3} = 1$ , what is the value of $x$ ?

Show answer & explanation

Answer: -3

Answer

The only real solution to the equation is

-3

.

The value

-3

is the only real number that satisfies the original equation. Substituting

-3

back into the original equation yields

\sqrt{2(-3) + 7} + \sqrt{-3 + 3} = \sqrt{1} + 0 = 1

, which is true.

Step-by-Step Solution

1

Isolate the first radical term.

\sqrt{2x + 7} = 1 - \sqrt{x + 3}

This allows for squaring both sides to eliminate one radical.

2

Square both sides and simplify.

2x + 7 = x + 4 - 2\sqrt{x + 3}

Squaring removes the radical on the left side, though it creates a middle term on the right side.

3

Isolate the remaining radical term.

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

Grouping the non-radical terms on one side prepares the equation for a second squaring step.

4

Square both sides again to eliminate the remaining radical.

x^2 + 6x + 9 = 4(x + 3)

Squaring both sides eliminates the radical completely, converting the expression into a polynomial equation.

5

Solve the quadratic equation.

x = -3

and

x = 1

Rearranging to

x^2 + 2x - 3 = 0

and factoring as

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

gives the candidate solutions.

6

Substitute candidates back into the original equation to check for extraneous solutions.

The only valid solution is

x = -3

.

Substituting

x = 1

yields

5 = 1

(invalid), while substituting

x = -3

yields

1 = 1

(valid).

Key Concept

Solving radical equations by isolating radicals and squaring, then testing for extraneous solutions.

Answer

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