Advanced Math Practice Questions — SAT — Page 3

A system of two equations is shown below.

y = x^2 - 4

y = x - 2

Which of the following ordered pairs $(x, y)$ is a solution to the system?

(2, 0)

(0, -2)

(-2, 0)

(1, -3)

Show answer & explanation

Answer: $(2, 0)$

Answer

The ordered pair

(2, 0)

is the correct solution to the system.

The ordered pair

(2, 0)

is the correct solution because substituting these values into both equations in the system results in true statements. For the quadratic equation,

0 = 2^2 - 4

simplifies to

0 = 0

. For the linear equation,

0 = 2 - 2

simplifies to

0 = 0

.

Step-by-Step Solution

1

Substitute the

x

-value and

y

-value from the candidate solution

(2, 0)

into the first equation,

y = x^2 - 4

.

0 = 2^2 - 4 \implies 0 = 4 - 4 \implies 0 = 0

, which is a true statement.

An ordered pair must satisfy the first equation to be a candidate solution for the system.

2

Substitute the

x

-value and

y

-value from the candidate solution

(2, 0)

into the second equation,

y = x - 2

.

0 = 2 - 2 \implies 0 = 0

, which is also a true statement.

An ordered pair must satisfy all equations in the system simultaneously to be a valid solution.

Key Concept

Verifying a solution to a nonlinear system of equations by substituting the coordinate values into both equations.

Question 42Question →

If the expression $\frac{6x^3 + 7x^2 - 14x + 19}{2x + 5}$ is equivalent to $ax^2 + bx + c + \frac{k}{2x + 5}$ for all $x \neq -2.5$ , where $a$ , $b$ , $c$ , and $k$ are constants, what is the value of $a - b + c + k$ ?

Show answer & explanation

Answer: 14

Answer

14

Dividing the numerator

6x^3 + 7x^2 - 14x + 19

by the denominator

2x + 5

using polynomial division yields a quotient of

3x^2 - 4x + 3

and a remainder of

4

. The equivalent expression is

3x^2 - 4x + 3 + \frac{4}{2x + 5}

. Comparing this with the form

ax^2 + bx + c + \frac{k}{2x + 5}

gives the values

a = 3

,

b = -4

,

c = 3

, and

k = 4

. Substituting these values into

a - b + c + k

gives

3 - (-4) + 3 + 4 = 14

.

Step-by-Step Solution

1

Perform the first step of polynomial long division by dividing

6x^3

by

2x

.

Quotient term:

3x^2

; Remainder:

-8x^2 - 14x + 19

To eliminate the highest-degree term of the numerator.

2

Perform the second step of division by dividing

-8x^2

by

2x

.

Quotient term:

-4x

; Remainder:

6x + 19

To find the next term of the quotient.

3

Perform the third step of division by dividing

6x

by

2x

.

Quotient term:

3

; Remainder:

4

To find the constant term of the quotient and the final remainder.

4

Compare the resulting expression

3x^2 - 4x + 3 + \frac{4}{2x + 5}

with the given form to identify the constants

a

,

b

,

c

, and

k

.

a = 3

,

b = -4

,

c = 3

,

k = 4

To map the coefficients of equivalent algebraic expressions.

5

Evaluate the expression

a - b + c + k

using the identified values.

3 - (-4) + 3 + 4 = 14

To obtain the final numeric value requested.

Key Concept

Equivalent Algebraic Expressions via Polynomial Long Division

Question 43Question →

For all positive values of $x$ , which of the following expressions is equivalent to $(8x^6)^{\frac{1}{3}}$ ?

2x^3

\frac{8}{3}x^2

2x^2

8x^2

Show answer & explanation

Answer: $2x^2$

Answer

The expression

2x^2

To find an equivalent expression, apply the exponent of

\frac{1}{3}

to each factor inside the parentheses. The expression

(8x^6)^{\frac{1}{3}}

becomes

8^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}}

. Since

8^{\frac{1}{3}} = \sqrt[3]{8} = 2

, and

(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2

, the simplified expression is

2x^2

.

Step-by-Step Solution

1

Apply the power of a product rule,

(ab)^n = a^n b^n

, to distribute the exponent of

\frac{1}{3}

to both the coefficient and the variable term.

(8x^6)^{\frac{1}{3}} = 8^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}}

This separates the numerical coefficient and the variable part to simplify them individually.

2

Simplify the numerical coefficient by evaluating the cube root of

8

.

8^{\frac{1}{3}} = 2

The exponent of

\frac{1}{3}

represents the cube root, and

2^3 = 8

.

3

Simplify the variable term using the power of a power rule,

(x^a)^b = x^{a \cdot b}

.

(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2

Multiplying the exponents simplifies the expression.

4

Multiply the simplified coefficient and variable term together.

2x^2

This gives the final simplified equivalent expression.

Key Concept

Simplifying algebraic expressions with fractional exponents using exponent rules.

Question 44Question →

For all real numbers $x$ and $y$ , the expression $(4x^3y^2)^2(2x^2y)$ is equivalent to which of the following?

8x^8y^5

32x^{12}y^4

32x^8y^5

32x^7y^5

Show answer & explanation

Answer: $32x^8y^5$

Answer

The correct equivalent expression is

32x^8y^5

.

To find the equivalent expression, we first apply the power of a product rule to the first term:

(4x^3y^2)^2 = 4^2(x^3)^2(y^2)^2 = 16x^6y^4

. Next, we multiply this result by the second term:

(16x^6y^4)(2x^2y)

. Multiplying the coefficients gives

16 \times 2 = 32

. Adding the exponents of the same base variables gives

x^{6+2} = x^8

and

y^{4+1} = y^5

. This results in

32x^8y^5

.

Step-by-Step Solution

1

Apply the power of a product rule to square the first expression:

(4x^3y^2)^2

.

16x^6y^4

When raising a product to a power, raise each factor to that power:

4^2 = 16

,

(x^3)^2 = x^{3 \times 2} = x^6

, and

(y^2)^2 = y^{2 \times 2} = y^4

.

2

Multiply the simplified first expression by the second expression:

(16x^6y^4)(2x^2y)

.

32x^8y^5

Multiply the numerical coefficients (

16 \times 2 = 32

) and add the exponents of variables with matching bases (

x^{6+2} = x^8

and

y^{4+1} = y^5

).

Key Concept

Simplifying equivalent algebraic expressions using rules of exponents

Estimated Time:45s

Question 45Question →

If the expression $\frac{2x + 1}{x - 3} - \frac{x - 2}{x + 1}$ is rewritten in the equivalent form $a + \frac{bx + c}{x^2 - 2x - 3}$ for all $x > 3$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

-1

11

9

19

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. Finding a common denominator yields the combined numerator

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

. Expanding these terms gives

(2x^2 + 3x + 1) - (x^2 - 5x + 6) = x^2 + 8x - 5

. Dividing

x^2 + 8x - 5

by

x^2 - 2x - 3

results in a quotient of

1

and a remainder of

10x - 2

. Thus, the expression is equivalent to

1 + \frac{10x - 2}{x^2 - 2x - 3}

, which gives

a = 1

,

b = 10

, and

c = -2

. The sum

a + b + c = 1 + 10 - 2 = 9

.

Step-by-Step Solution

1

Find a common denominator for the two rational expressions.

The common denominator is

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

. The combined expression is

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

.

To combine the fractions, we need to express them with a common denominator.

2

Expand and simplify the numerator.

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

and

(x - 2)(x - 3) = x^2 - 5x + 6

. Subtracting them gives

(2x^2 + 3x + 1) - (x^2 - 5x + 6) = x^2 + 8x - 5

.

Simplifying the numerator allows us to express the combined fraction as a single polynomial over the denominator.

3

Perform polynomial division or rewrite the numerator to match the form

a + \frac{bx + c}{x^2 - 2x - 3}

.

Rewriting the numerator:

x^2 + 8x - 5 = 1(x^2 - 2x - 3) + 10x - 2

. Thus, the expression becomes

1 + \frac{10x - 2}{x^2 - 2x - 3}

.

This separates the rational expression into a constant integer and a proper rational fraction.

4

Identify the values of

a

,

b

, and

c

, and find their sum.

a = 1

,

b = 10

, and

c = -2

. The sum is

a + b + c = 1 + 10 + (-2) = 9

.

We compare the coefficients from our result to the given form and calculate the requested sum.

Key Concept

Combining rational expressions and rewriting them using polynomial division or algebraic manipulation.

Estimated Time:2m 30s

Question 46Question →

If $(x, y)$ is a solution to the system of equations below and $x > 0$ , what is the value of $x$ ?

y = x^2

y = x + 6

Show answer & explanation

Answer: 3

Answer

The correct answer is

3

.

Substituting

y = x^2

into

y = x + 6

gives the quadratic equation

x^2 - x - 6 = 0

. Factoring this expression yields

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

, which gives solutions of

x = 3

and

x = -2

. Since the system requires

x > 0

, the only valid solution is

3

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

x^2 = x + 6

To eliminate the variable

y

and solve for

x

directly.

2

Subtract

x

and

6

from both sides to write the quadratic equation in standard form.

x^2 - x - 6 = 0

Setting the quadratic expression equal to zero allows it to be factored.

3

Factor the quadratic trinomial.

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

Finding factors whose product is

-6

and whose sum is

-1

helps find the roots.

4

Solve for

x

and apply the constraint

x > 0

.

x = 3

The equation has solutions

x = 3

and

x = -2

. Because

x

must be greater than

0

, we discard the negative solution.

Key Concept

Solving a system of nonlinear equations by substitution and factoring the resulting quadratic equation.

Question 47Question →

If the expression $5(2x - 3) - 4(x - 2)$ is equivalent to $ax + b$ for all values of $x$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Show answer & explanation

Answer: 6

Answer

The value of the constant coefficient is 6.

Expanding the expression

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

yields

10x - 15 - 4x + 8

. Combining the like terms results in

(10x - 4x) + (-15 + 8) = 6x - 7

. Comparing this to the expression

ax + b

, the constant coefficient

a

is equal to 6.

Step-by-Step Solution

1

Distribute the multipliers to the terms inside the parentheses.

10x - 15 - 4x + 8

To remove the parentheses and prepare the expression for simplification, multiply each term inside

(2x - 3)

by

5

and each term inside

(x - 2)

by

-4

.

2

Combine the linear terms and the constant terms.

6x - 7

Combine the variable terms (

10x - 4x = 6x

) and the constants (

-15 + 8 = -7

) to rewrite the expression in its simplest form.

3

Compare the simplified expression to the form

ax + b

to identify the value of

a

.

a = 6

The coefficient of the variable

x

in

6x - 7

corresponds directly to

a

in

ax + b

.

Key Concept

Equivalent Algebraic Expressions

Question 48Question →

For all positive real numbers $x$ and $y$ , the expression $\frac{(x^{3} y^{2})^{2/3} \cdot (x^{-1} y^{4})^{1/6}}{(x^2 y)^{1/2}}$ can be written in the equivalent form $x^a y^b$ , where $a$ and $b$ are constants. What is the value of $6a + 2b$ ?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

Applying the rules of exponents systematically yields the simplified expression

x^{5/6} y^{3/2}

. By setting

a = \frac{5}{6}

and

b = \frac{3}{2}

, the linear combination

6a + 2b

evaluates to

6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

.

Step-by-Step Solution

1

Apply the power of a power rule to the first term in the numerator.

(x^3 y^2)^{2/3} = x^{3 \cdot \frac{2}{3}} y^{2 \cdot \frac{2}{3}} = x^2 y^{4/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

2

Apply the power of a power rule to the second term in the numerator.

(x^{-1} y^4)^{1/6} = x^{-1 \cdot \frac{1}{6}} y^{4 \cdot \frac{1}{6}} = x^{-1/6} y^{2/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

3

Multiply the two simplified terms in the numerator.

(x^2 y^{4/3})(x^{-1/6} y^{2/3}) = x^{2 - \frac{1}{6}} y^{\frac{4}{3} + \frac{2}{3}} = x^{11/6} y^2

When multiplying expressions with the same base, add their exponents.

4

Simplify the denominator.

(x^2 y)^{1/2} = x^{2 \cdot \frac{1}{2}} y^{1 \cdot \frac{1}{2}} = x y^{1/2}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

5

Divide the numerator by the denominator.

\frac{x^{11/6} y^2}{x y^{1/2}} = x^{\frac{11}{6} - 1} y^{2 - \frac{1}{2}} = x^{5/6} y^{3/2}

When dividing expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

6

Identify the values of

a

and

b

and calculate

6a + 2b

.

a = \frac{5}{6}

,

b = \frac{3}{2}

, so

6a + 2b = 6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

Matching the simplified expression

x^{5/6} y^{3/2}

to

x^a y^b

yields the values of the constants

a

and

b

, which are then used to calculate the required expression.

Key Concept

Simplifying rational expressions with fractional exponents using exponent rules.

Question 49Question →

If the expression $(x + 5)^2 - (x - 3)^2$ is equivalent to $ax + b$ for all values of $x$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Show answer & explanation

Answer: 16

Answer

16

Expanding

(x + 5)^2

gives

x^2 + 10x + 25

, and expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

x^2 + 10x + 25 - (x^2 - 6x + 9) = x^2 + 10x + 25 - x^2 + 6x - 9

. Combining like terms yields

16x + 16

. Comparing this to

ax + b

shows that the coefficient of

x

,

a

, is 16.

Step-by-Step Solution

1

Expand the first squared binomial term

(x + 5)^2 = x^2 + 10x + 25

To express the binomial square as a trinomial using the perfect square identity

(u + v)^2 = u^2 + 2uv + v^2

.

2

Expand the second squared binomial term

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

To express the binomial square as a trinomial using the perfect square identity

(u - v)^2 = u^2 - 2uv + v^2

.

3

Subtract the expanded expressions and distribute the negative sign

x^2 + 10x + 25 - x^2 + 6x - 9

To combine the terms while correctly applying the distributive property to the subtracted expression.

4

Combine like terms to simplify the polynomial

16x + 16

To find the final simplified polynomial of the form

ax + b

.

5

Compare the simplified expression to the standard form to find the value of

a

a = 16

The constant

a

represents the coefficient of the linear term

x

, which is 16.

Key Concept

Simplifying algebraic expressions by expanding binomial products and combining like terms.

Question 50Question →

For all positive real numbers $a$ and $b$ such that $a \neq b$ , which of the following is equivalent to the expression $\frac{a^{3/2} - b^{3/2}}{\sqrt{a} - \sqrt{b}}$ ?

a + \sqrt{ab} + b

a + b

a - \sqrt{ab} + b

a + 2\sqrt{ab} + b

Show answer & explanation

Answer: $a + \sqrt{ab} + b$

Answer

a + \sqrt{ab} + b

The correct answer is obtained by expressing the numerator as a difference of cubes:

(\sqrt{a})^3 - (\sqrt{b})^3

. Factoring this expression gives

(\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)

. Since

a \neq b

, dividing by the denominator

\sqrt{a} - \sqrt{b}

simplifies the expression to

a + \sqrt{ab} + b

.

Step-by-Step Solution

1

Rewrite the terms in the numerator using square roots to reveal a difference of cubes pattern.

a^{3/2} = (\sqrt{a})^3

and

b^{3/2} = (\sqrt{b})^3

, so the numerator is

(\sqrt{a})^3 - (\sqrt{b})^3

.

This allows us to factor the numerator using the algebraic identity for the difference of two cubes.

2

Factor the numerator using the difference of cubes formula:

u^3 - v^3 = (u - v)(u^2 + uv + v^2)

.

(\sqrt{a})^3 - (\sqrt{b})^3 = (\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)

where

u = \sqrt{a}

and

v = \sqrt{b}

.

Factoring allows us to identify common factors shared with the denominator.

3

Substitute the factored expression back into the fraction and cancel the common factor of

\sqrt{a} - \sqrt{b}

.

\frac{(\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)}{\sqrt{a} - \sqrt{b}} = a + \sqrt{ab} + b

.

Since

a \neq b

,

\sqrt{a} - \sqrt{b} \neq 0

, which makes it mathematically valid to divide by this term.

Key Concept

Equivalent algebraic expressions involving fractional exponents and difference of cubes factoring

Alternative Method

Let

a = 4

and

b = 1

. Substitute these values into the original expression:

\frac{4^{3/2} - 1^{3/2}}{\sqrt{4} - \sqrt{1}} = \frac{8 - 1}{2 - 1} = 7

. Now substitute these same values into each option to see which one evaluates to 7. The correct option evaluates to

4 + \sqrt{4(1)} + 1 = 7

.

Estimated Time:2m 0s

Question 51Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{2x^2 - 5x - 3}{2x + 1} - (x - 4)$ ?

-7

2x - 7

1

7

Show answer & explanation

Answer: 1

Answer

1

The correct answer is

1

. Factoring the numerator of the rational expression gives

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

. Since

x > 1

, the denominator

2x + 1

is non-zero, allowing the expression to be simplified to

x - 3

. Subtracting

(x - 4)

and distributing the negative sign to both terms inside the parentheses yields

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

.

Step-by-Step Solution

1

Factor the quadratic expression in the numerator of the rational term.

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

This allows for the identification of common factors that can be simplified with the denominator.

2

Simplify the rational expression by canceling the common factor in the numerator and denominator.

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

Since

x > 1

, the term

2x + 1

is positive and non-zero, making the division valid.

3

Subtract the linear expression from the simplified rational expression, distributing the negative sign to both terms inside the parentheses.

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

This performs the final subtraction and simplifies the expression to its equivalent constant form.

Key Concept

Simplifying rational expressions by factoring and performing operations on equivalent expressions

Estimated Time:1m 15s

Question 52Question →

The graphs of the equations $y = x^2 - 10$ and $y = 2x - 2$ intersect at the point $(x, y)$ in the first quadrant. What is the value of $y$ ?

Show answer & explanation

Answer: 6

Answer

The value of

y

is 6.

Equating the equations gives

x^2 - 10 = 2x - 2

. Moving all terms to one side yields

x^2 - 2x - 8 = 0

, which factors as

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

. Since the point is in the first quadrant, both coordinates must be positive, so we use

x = 4

. Substituting

x = 4

into the linear equation gives

y = 2(4) - 2 = 6

.

Step-by-Step Solution

1

Equate the two expressions for

y

x^2 - 10 = 2x - 2

Since both equations define

y

, their right-hand sides must be equal at the points of intersection.

2

Rewrite the equation in standard quadratic form

ax^2 + bx + c = 0

x^2 - 2x - 8 = 0

Subtract

2x

and add

2

to both sides of the equation to set it equal to zero.

3

Factor the quadratic equation

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

Find two integers that multiply to

-8

and add to

-2

, which are

-4

and

2

.

4

Solve for the possible values of

x

x = 4

or

x = -2

Set each factor equal to zero and solve.

5

Determine the positive

x

-coordinate and find

y

y = 6

For the point to be in the first quadrant, both coordinates must be positive. Thus, we select

x = 4

and substitute it into the linear equation:

y = 2(4) - 2 = 6

.

Key Concept

Solving a system of linear and quadratic equations via substitution.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

x^2 - 2x - 8 = 0

:

x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2}

. This yields

x = 4

and

x = -2

. Then substitute the positive root to find

y

.

Estimated Time:1m 30s

Question 53Question →

If the expression $\frac{3x^2 + 10x - 8}{x + k}$ is equivalent to $3x - 2$ for all $x \neq -k$ , where $k$ is a positive constant, what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

k

is 4.

For the expressions to be equivalent for all values of

x

, the numerator

3x^2 + 10x - 8

must be equal to the product of the denominator

x + k

and the simplified quotient

3x - 2

. Expanding the product yields

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

. Equating the constant terms on both sides gives

-8 = -2k

, which results in

k = 4

. Alternatively, equating the coefficients of the linear terms gives

10 = 3k - 2

, which also yields

k = 4

.

Step-by-Step Solution

1

Multiply both sides of the equivalence by the denominator

x + k

.

3x^2 + 10x - 8 = (3x - 2)(x + k)

To eliminate the fraction and set up a polynomial identity.

2

Expand the right side of the equation.

3x^2 + 10x - 8 = 3x^2 + (3k - 2)x - 2k

To express the right side in standard quadratic form for coefficient comparison.

3

Equate the constant terms to solve for

k

.

-8 = -2k \implies k = 4

Since the expressions are equivalent for all values of

x

, their corresponding coefficients and constants must be equal.

Key Concept

Equating coefficients of equivalent polynomial expressions

Question 54Question →

If the expression $\frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1}$ is equivalent to $ax + b + \frac{cx + d}{2x^2 + 1}$ for all values of $x$ , where $a$ , $b$ , $c$ , and $d$ are constants, what is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 14

Answer

The value of

a + b + c + d

is 14.

Performing polynomial long division on

\frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1}

yields a quotient of

2x - 1

and a remainder of

5x + 8

. Matching this to the form

ax + b + \frac{cx + d}{2x^2 + 1}

gives

a = 2

,

b = -1

,

c = 5

, and

d = 8

. Summing these values gives

2 + (-1) + 5 + 8 = 14

.

Step-by-Step Solution

1

Divide the leading term of the numerator by the leading term of the denominator to find the first term of the quotient.

\frac{4x^3}{2x^2} = 2x

. Multiplying

2x(2x^2 + 1) = 4x^3 + 2x

. Subtracting this from the numerator yields

-2x^2 + 5x + 7

.

To initiate the polynomial division process.

2

Divide the leading term of the remaining polynomial by the leading term of the divisor to find the second term of the quotient.

\frac{-2x^2}{2x^2} = -1

. Multiplying

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

. Subtracting this from the remaining polynomial yields

5x + 8

.

To find the next term of the quotient and determine the remainder.

3

Write the expression in the quotient-remainder form and identify the values of the constants

a

,

b

,

c

, and

d

.

The expression is equivalent to

2x - 1 + \frac{5x + 8}{2x^2 + 1}

, so

a = 2

,

b = -1

,

c = 5

, and

d = 8

.

To match the given algebraic form of the expression.

4

Calculate the sum of the constants

a + b + c + d

.

2 + (-1) + 5 + 8 = 14

.

To find the final value requested by the question.

Key Concept

Equivalent Algebraic Expressions

Question 55Question →

For all $x > 8$ , the expression $\frac{x^{8/3} - 8x^{5/3}}{x^{4/3} - 4x^{2/3}} \cdot \frac{x^{1/3} + 2}{x^{4/3} + 2x + 4x^{2/3}}$ is equivalent to $x^a$ , where $a$ is a constant. What is the value of $\frac{1}{a}$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Factoring the numerator and denominator of the first fraction yields

\frac{x^{5/3}(x^{1/3}-2)(x^{2/3}+2x^{1/3}+4)}{x^{2/3}(x^{1/3}-2)(x^{1/3}+2)}

, which simplifies to

\frac{x(x^{2/3}+2x^{1/3}+4)}{x^{1/3}+2}

. Factoring the denominator of the second fraction yields

\frac{x^{1/3}+2}{x^{2/3}(x^{2/3}+2x^{1/3}+4)}

. Multiplying these two simplified expressions cancels the common terms

(x^{1/3}+2)

and

(x^{2/3}+2x^{1/3}+4)

, leaving

\frac{x}{x^{2/3}} = x^{1/3}

. Therefore,

a = \frac{1}{3}

, and the value of the reciprocal

\frac{1}{a}

is

3

.

Step-by-Step Solution

1

Factor the numerator and the denominator of the first fraction.

The first fraction becomes

\frac{x(x^{2/3} + 2x^{1/3} + 4)}{x^{1/3} + 2}

.

Factoring out

x^{5/3}

from the numerator gives

x^{5/3}(x-8)

, and factoring out

x^{2/3}

from the denominator gives

x^{2/3}(x^{2/3}-4)

. Using the difference of cubes

x - 8 = (x^{1/3} - 2)(x^{2/3} + 2x^{1/3} + 4)

and the difference of squares

x^{2/3} - 4 = (x^{1/3} - 2)(x^{1/3} + 2)

, we can cancel the common factor

(x^{1/3} - 2)

.

2

Factor the denominator of the second fraction.

The second fraction becomes

\frac{x^{1/3} + 2}{x^{2/3}(x^{2/3} + 2x^{1/3} + 4)}

.

Factoring out

x^{2/3}

from the expression

x^{4/3} + 2x + 4x^{2/3}

reveals a common quadratic-like term

(x^{2/3} + 2x^{1/3} + 4)

that can be used for cancellation.

3

Multiply the two rational expressions together and simplify.

x^{1/3}

Multiplying the simplified fractions allows us to cancel the common binomial term

(x^{1/3} + 2)

and the trinomial term

(x^{2/3} + 2x^{1/3} + 4)

, leaving

\frac{x}{x^{2/3}} = x^{1 - 2/3} = x^{1/3}

.

4

Find the value of

\frac{1}{a}

.

3

Since the expression is equivalent to

x^a

, we identify

a = \frac{1}{3}

. Taking the reciprocal of

a

gives

\frac{1}{1/3} = 3

.

Key Concept

Equivalent Algebraic Expressions

Question 56Question →

The expression $(3x^2 + 5x) - (x^2 - kx)$ , where $k$ is a constant, can be rewritten as $2x^2 + 12x$ . What is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

To find the value of

k

, we first simplify the expression

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

by distributing the subtraction sign to both terms inside the second set of parentheses. This yields

3x^2 + 5x - x^2 + kx

. Next, we group and combine like terms to get

(3x^2 - x^2) + (5x + kx) = 2x^2 + (5+k)x

. Since this expression is equivalent to

2x^2 + 12x

for all values of

x

, the coefficients of corresponding terms must be equal. Equating the coefficients of

x

gives

5+k = 12

. Subtracting 5 from both sides yields

k = 7

.

Step-by-Step Solution

1

Distribute the negative sign to the terms in the second parentheses.

3x^2 + 5x - x^2 + kx

To remove the parentheses and simplify the expression.

2

Combine like terms.

2x^2 + (5 + k)x

Grouping the

x^2

terms and

x

terms simplifies comparison with the target expression.

3

Equate the coefficient of the

x

term to the corresponding coefficient in the target expression.

5 + k = 12 \implies k = 7

Equivalent expressions must have equal corresponding coefficients for all values of

x

.

Key Concept

Equivalence of polynomial expressions by combining like terms and equating coefficients

Question 57Question →

A system of equations is shown below.

y = x^2 - 2x

y = 3

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

2

3

6

4

Show answer & explanation

Answer: 6

Answer

6

To solve the system of equations, substitute the expression for

y

from the second equation into the first equation:

3 = x^2 - 2x

. Subtracting 3 from both sides results in the quadratic equation

x^2 - 2x - 3 = 0

. Factoring the quadratic yields

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

, which gives the possible values of

x

as

3

and

-1

. Since the problem specifies that

x > 0

, the value of

x

is 3. Given that

y = 3

, the value of

x + y

is

3 + 3 = 6

.

Step-by-Step Solution

1

Substitute the value of

y

from the second equation into the first equation.

3 = x^2 - 2x

Since both equations are equal to

y

, their right-hand sides must be equal to each other.

2

Rearrange the equation to set it equal to zero.

x^2 - 2x - 3 = 0

This puts the equation into standard quadratic form so that it can be factored.

3

Factor the quadratic equation.

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

Finding factors of -3 that add up to -2 helps isolate the solutions for

x

.

4

Find the solutions for

x

and apply the given constraint.

x = 3

(since

x > 0

)

The factors give

x = 3

and

x = -1

. The constraint

x > 0

excludes

x = -1

.

5

Calculate the value of

x + y

.

3 + 3 = 6

Substitute the value of

x = 3

and the given value of

y = 3

to find the final sum.

Key Concept

Solving nonlinear systems of equations using substitution and solving quadratic equations by factoring.

Question 58Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{x^2 - \sqrt{x}}{x - \sqrt{x}} - \sqrt{x}$ ?

x - 2\sqrt{x} + 1

x - \sqrt{x} - 1

x + 1

x - 1

Show answer & explanation

Answer: $x + 1$

Answer

The expression is equivalent to

x + 1

.

The expression can be simplified by substituting

u = \sqrt{x}

, which gives

x = u^2

and

x^2 = u^4

. Substituting these into the original expression yields

\frac{u^4 - u}{u^2 - u} - u

. Factoring out

u

from the numerator and denominator gives

\frac{u(u^3 - 1)}{u(u - 1)} - u = \frac{u^3 - 1}{u - 1} - u

. Factoring the difference of cubes in the numerator as

(u - 1)(u^2 + u + 1)

and canceling the common factor of

u - 1

leaves

u^2 + u + 1 - u = u^2 + 1

. Substituting back

x = u^2

yields the equivalent expression

x + 1

.

Step-by-Step Solution

1

Substitute

u = \sqrt{x}

into the expression, which implies

x = u^2

and

x^2 = u^4

.

The expression becomes

\frac{u^4 - u}{u^2 - u} - u

.

Using a substitution simplifies the fractional exponents and makes the polynomial structure easier to recognize.

2

Factor out

u

from both the numerator and the denominator of the fraction.

\frac{u(u^3 - 1)}{u(u - 1)} - u = \frac{u^3 - 1}{u - 1} - u

.

Since

x > 1

, we have

u > 1

, which means

u \neq 0

. Therefore, we can cancel the common factor

u

from the numerator and denominator.

3

Factor the difference of cubes in the numerator:

u^3 - 1 = (u - 1)(u^2 + u + 1)

.

\frac{(u - 1)(u^2 + u + 1)}{u - 1} - u

.

Factoring the numerator allows us to simplify the rational expression by canceling the common binomial factor in the denominator.

4

Cancel the common factor

u - 1

and simplify the remaining terms.

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

.

Since

u > 1

, we have

u - 1 \neq 0

, allowing us to divide out

u - 1

. Subtracting

u

from

u^2 + u + 1

leaves

u^2 + 1

.

5

Substitute

x

back in place of

u^2

.

x + 1

.

Converting the simplified expression back to the original variable gives the final equivalent algebraic expression.

Key Concept

Simplifying rational expressions with fractional exponents by substitution and factoring.

Question 59Question →

For all $x > 4$ , the expression $\frac{x^2 - 4x}{x - 2\sqrt{x}} + \frac{x\sqrt{x} + 8}{\sqrt{x} + 2}$ can be written in the form $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 6

Answer

6

Factoring the numerator of the first term yields

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

and its denominator yields

\sqrt{x}(\sqrt{x}-2)

. Simplifying this term gives

x + 2\sqrt{x}

. Factoring the numerator of the second term as a sum of cubes gives

(\sqrt{x}+2)(x - 2\sqrt{x} + 4)

, which simplifies to

x - 2\sqrt{x} + 4

. Summing both simplified terms results in

2x + 4

. Matching this to the form

ax+b

gives

a=2

and

b=4

, so

a+b=6

.

Step-by-Step Solution

1

Simplify the first term of the expression.

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

Factor

x

from the numerator to get

x(x-4)

and

\sqrt{x}

from the denominator to get

\sqrt{x}(\sqrt{x}-2)

. Rewrite

x-4

as the difference of squares

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

, then cancel the common factor

\sqrt{x}-2

and simplify

\frac{x}{\sqrt{x}}

to

\sqrt{x}

.

2

Simplify the second term of the expression.

\frac{x\sqrt{x} + 8}{\sqrt{x} + 2} = x - 2\sqrt{x} + 4

Recognize

x\sqrt{x} + 8

as a sum of cubes,

(\sqrt{x})^3 + 2^3

. Factor it as

(\sqrt{x}+2)(x - 2\sqrt{x} + 4)

and cancel the common factor of

\sqrt{x}+2

in the denominator.

3

Add the simplified terms together.

2x + 4

Combine

(x + 2\sqrt{x})

and

(x - 2\sqrt{x} + 4)

by grouping like terms: the

2\sqrt{x}

and

-2\sqrt{x}

cancel out, leaving

2x + 4

.

4

Identify the values of

a

and

b

and find

a+b

.

6

Comparing

2x + 4

to

ax + b

gives

a = 2

and

b = 4

. Therefore,

a + b = 2 + 4 = 6

.

Key Concept

Simplifying rational expressions involving radicals by factoring (difference of squares and sum of cubes).

Question 60Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{4x^2 - 1}{2x - 1} - \frac{6x^2 + x - 2}{3x + 2}$ ?

0

2

-2

4x

Show answer & explanation

Answer: 2

Answer

2

Factoring the numerators allows the expression to be simplified. The first term becomes

2x + 1

, and the second term becomes

2x - 1

. Subtracting the second term from the first and distributing the negative sign results in

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

.

Step-by-Step Solution

1

Factor the numerator of the first rational term,

4x^2 - 1

, using the difference of squares identity.

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

To identify and divide out common factors between the numerator and denominator.

2

Simplify the first term by dividing the factored numerator by its denominator,

2x - 1

.

\frac{(2x - 1)(2x + 1)}{2x - 1} = 2x + 1

Since

x > 1

,

2x - 1 \neq 0

, allowing the division.

3

Factor the numerator of the second rational term,

6x^2 + x - 2

.

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

To find common factors that can be simplified with the denominator.

4

Simplify the second term by dividing the factored numerator by its denominator,

3x + 2

.

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

Since

x > 1

,

3x + 2 \neq 0

, allowing the division.

5

Subtract the second simplified expression from the first, ensuring that the negative sign is correctly distributed to all terms.

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

To combine the terms and find the final equivalent value of the entire expression.

Key Concept

Equivalent Algebraic Expressions

Advanced Math

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