Algebra and Functions Practice Questions — GMAT — Page 4

Question 61Question →

For all real numbers $x$ and $y$ , the custom operation $\odot$ is defined by $x \odot y = x^2 - 2y$ . What is the value of $(3 \odot 2) \odot 4$ ?

17

6

-4

9

-7

Show answer & explanation

Answer: 17

Answer

17

Following the explicit operator rule

x \odot y = x^2 - 2y

, first calculate the inner expression

(3 \odot 2) = 3^2 - 2(2) = 5

. Next, substitute 5 into the outer expression to compute

5 \odot 4 = 5^2 - 2(4) = 25 - 8 = 17

. Thus, the value 17 is correct.

Step-by-Step Solution

1

Evaluate the inner function expression

(3 \odot 2)

using the rule

x \odot y = x^2 - 2y

3 \odot 2 = 3^2 - 2(2) = 9 - 4 = 5

Parentheses dictate that inner custom operations must be evaluated first.

2

Substitute the result of step 1 back into the expression to evaluate

5 \odot 4

5 \odot 4 = 5^2 - 2(4) = 25 - 8 = 17

Applying the custom operator rule with

x = 5

and

y = 4

computes the final target value.

Key Concept

Custom Operator Evaluation and Order of Operations

Estimated Time:1m 15s

Question 62Question →

If $x(x^2 - 7x + 12) = 2(x^2 - 7x + 12)$ , what is the sum of all distinct real values of $x$ that satisfy the equation?

2

5

7

9

14

Show answer & explanation

Answer: 9

Answer

9

To find all solutions to

x(x^2 - 7x + 12) = 2(x^2 - 7x + 12)

, subtract the right side from the left to obtain

(x - 2)(x^2 - 7x + 12) = 0

. Factoring the quadratic expression gives

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

. Setting each factor to zero reveals three distinct real roots: 2, 3, and 4. The sum of these roots is

2 + 3 + 4 = 9

.

Step-by-Step Solution

1

Rearrange the equation to set one side to zero

x(x^2 - 7x + 12) - 2(x^2 - 7x + 12) = 0

Setting the equation to zero allows the application of the zero-product property without illegally dividing by variable expressions.

2

Factor out the common quadratic expression

(x^2 - 7x + 12)

(x - 2)(x^2 - 7x + 12) = 0

Both terms share the common factor

(x^2 - 7x + 12)

.

3

Factor the quadratic component

(x^2 - 7x + 12)

into linear factors

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

The quadratic factors into

(x - 3)(x - 4)

because

(-3) \times (-4) = 12

and

(-3) + (-4) = -7

.

4

Apply the zero-product property to find all distinct real roots and compute their sum

Roots are

x = 2

,

x = 3

, and

x = 4

. Sum =

2 + 3 + 4 = 9

.

Setting each linear factor to zero gives the solutions. Summing these solutions yields 9.

Key Concept

Polynomial Factoring and the Zero-Product Property

Estimated Time:2m 0s

Question 63Question →

What is the sum of all distinct real values of $x$ that satisfy the equation $x(x^2 - 5x + 6) = 2(x^2 - 5x + 6)$ ?

2

3

5

7

9

Show answer & explanation

Answer: 5

Answer

The sum of all distinct real solutions is 5.

To solve the equation

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

, move all terms to the left side to get

x(x^2 - 5x + 6) - 2(x^2 - 5x + 6) = 0

. Factoring out

(x^2 - 5x + 6)

gives

(x - 2)(x^2 - 5x + 6) = 0

. Factoring the quadratic part yields

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

, or

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

. The real roots are

x = 2

and

x = 3

. Taking the sum of these distinct real roots gives

2 + 3 = 5

.

Step-by-Step Solution

1

Move all terms to one side of the equation to avoid dividing by a variable expression.

x(x^2 - 5x + 6) - 2(x^2 - 5x + 6) = 0

Dividing by

(x^2 - 5x + 6)

directly would lose valid roots where

x^2 - 5x + 6 = 0

.

2

Factor out the common quadratic expression

(x^2 - 5x + 6)

.

(x - 2)(x^2 - 5x + 6) = 0

Applying the distributive property isolates the linear factor

(x - 2)

.

3

Factor the quadratic term

x^2 - 5x + 6

into linear factors.

(x - 2)(x - 2)(x - 3) = 0 \implies (x - 2)^2(x - 3) = 0

The quadratic factors into

(x - 2)(x - 3)

because

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

and

(-2) \times (-3) = 6

.

4

Identify all distinct real solutions and compute their sum.

Real solutions are

x = 2

and

x = 3

. Distinct sum:

2 + 3 = 5

.

The root

x = 2

has multiplicity 2, but only distinct values must be summed.

Key Concept

Polynomial Factoring and Variable Cancellation Rules

Estimated Time:1m 30s

Question 64Question →

If $x$ is a real number such that $(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1$ , what is the sum of all distinct real values of $x$ that satisfy the equation?

10

12

14

15

9

Show answer & explanation

Answer: 15

Answer

The sum of all distinct real values of

x

that satisfy the equation is 15.

The equation

(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1

holds under three distinct conditions: when the exponent is 0 and the base is non-zero (

x = 4, 5

), when the base is 1 (

x = 1, 4

), and when the base is

-1

provided the exponent is an even integer (

x = 2, 3

, yielding even exponents 6 and 2 respectively). The set of distinct solutions is

\{1, 2, 3, 4, 5\}

, and their sum is 15.

Step-by-Step Solution

1

Analyze the general algebraic conditions under which an expression of the form

u^v = 1

holds for real numbers

u

and

v

.

The equation

u^v = 1

is satisfied in three distinct cases: (1)

v = 0

and

u \neq 0

, (2)

u = 1

, or (3)

u = -1

with

v

being an even integer.

Any non-zero real number raised to the power of 0 equals 1, 1 raised to any power equals 1, and

-1

raised to an even integer power equals 1.

2

Evaluate Case 1 where the exponent

x^2 - 9x + 20 = 0

.

Factoring gives

(x - 4)(x - 5) = 0

, yielding

x = 4

and

x = 5

. Testing the base

x^2 - 5x + 5

at

x = 4

gives

1 \neq 0

, and at

x = 5

gives

5 \neq 0

. Thus

x = 4

and

x = 5

are valid solutions.

Exponent equal to zero produces

u^0 = 1

as long as the base

u \neq 0

.

3

Evaluate Case 2 where the base

x^2 - 5x + 5 = 1

.

Subtracting 1 gives

x^2 - 5x + 4 = 0

, which factors as

(x - 1)(x - 4) = 0

, yielding

x = 1

and

x = 4

. Both are valid.

Base equal to 1 raised to any real exponent equals 1.

4

Evaluate Case 3 where the base

x^2 - 5x + 5 = -1

and check if the exponent is an even integer.

Adding 1 gives

x^2 - 5x + 6 = 0

, which factors as

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

, yielding

x = 2

and

x = 3

. Testing

x = 2

in the exponent gives

2^2 - 9(2) + 20 = 6

(even). Testing

x = 3

gives

3^2 - 9(3) + 20 = 2

(even). Thus

x = 2

and

x = 3

are also valid solutions.

Base

-1

raised to an even integer power equals 1.

5

Combine all unique real values of

x

and compute their sum.

The set of distinct real solutions is

\{1, 2, 3, 4, 5\}

. Their sum is

1 + 2 + 3 + 4 + 5 = 15

.

Summing all identified distinct valid roots.

Key Concept

Solving polynomial exponential equations of the form

f(x)^{g(x)} = 1

by systematically testing base-exponent cases.

Estimated Time:2m 0s

Question 65Question →

For all real numbers

x

except

0

and

1

, the function

f

satisfies the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x + 3

What is the value of

f(2)

?

Show answer & explanation

Answer: 7

Answer

The value of

f(2)

is

7

.

Evaluating the functional equation iteratively at the points of the cycle

x = 2

,

x = -1

, and

x = \frac{1}{2}

produces a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this linear system gives

f(2) = 7

.

Step-by-Step Solution

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) + 3 \implies f(2) + 2f(-1) = 21

This establishes a relationship between

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) + 3 \implies f(-1) + 2f\left(\frac{1}{2}\right) = -6

This links

f(-1)

to

f\left(\frac{1}{2}\right)

to build a closed system of transformations under the mapping

g(x) = \frac{1}{1-x}

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) + 3 \implies f\left(\frac{1}{2}\right) + 2f(2) = \frac{15}{2}

This completes the cycle since

g\left(\frac{1}{2}\right) = 2

, creating a system of three linear equations with three unknown values.

4

Solve the system of three linear equations for

f(2)

Let

a = f(2)

,

b = f(-1)

, and

c = f\left(\frac{1}{2}\right)

. From equation (3),

c = \frac{15}{2} - 2a

. Substituting

c

into equation (2) gives

b + 2\left(\frac{15}{2} - 2a\right) = -6 \implies b = 4a - 21

. Substituting

b

into equation (1) gives

a + 2(4a - 21) = 21 \implies 9a - 42 = 21 \implies 9a = 63 \implies a = 7

.

Algebraic substitution yields the exact value of

f(2)

.

Key Concept

Cyclic Functional Equations and System Substitution

Estimated Time:2m 30s

Question 66Question →

If $x$ is a real number such that $x^2 - 5x + 3 = 0$ , what is the value of $x^2 + \frac{9}{x^2}$ ?

Show answer & explanation

Answer: 19

Answer

The value of

x^2 + \frac{9}{x^2}

is 19.

Dividing

x^2 - 5x + 3 = 0

by

x

gives

x + \frac{3}{x} = 5

. Squaring both sides yields

\left(x + \frac{3}{x}\right)^2 = x^2 + 2(x)\left(\frac{3}{x}\right) + \frac{9}{x^2} = 25

, which simplifies to

x^2 + 6 + \frac{9}{x^2} = 25

. Subtracting 6 from both sides yields the final answer of 19.

Step-by-Step Solution

1

Divide the quadratic equation by

x

x + \frac{3}{x} = 5

Since

x = 0

is not a root of

x^2 - 5x + 3 = 0

, dividing by

x

rearranges the terms into

x

and its reciprocal component.

2

Square both sides of

x + \frac{3}{x} = 5

x^2 + 6 + \frac{9}{x^2} = 25

Applying the algebraic identity

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

generates the required expression

x^2 + \frac{9}{x^2}

alongside a constant middle term.

3

Isolate

x^2 + \frac{9}{x^2}

x^2 + \frac{9}{x^2} = 19

Subtracting the constant cross-term 6 from 25 yields the target value.

Key Concept

Algebraic manipulation of quadratic equations using reciprocal square identities

Estimated Time:2m 0s

Question 67Question →

What is the sum of all real solutions to the polynomial equation

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

Show answer & explanation

Answer: 6

Answer

The sum of all real solutions to the equation is 6.

The equation

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

represents

f(f(x)) = x

for

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

. Expanding and setting to zero yields

x^4 - 6x^3 + 8x^2 + 2x - 1 = 0

. Since solutions to

f(x) = x

(i.e.,

x^2 - 4x + 1 = 0

) also satisfy

f(f(x)) = x

, we can factor out

x^2 - 4x + 1

to get

(x^2 - 4x + 1)(x^2 - 2x - 1) = 0

. Both quadratic factors have positive discriminants (

12

and

8

), ensuring four distinct real roots. By Vieta's formulas, the sum of roots from the first quadratic is 4 and from the second is 2, giving a total sum of 6.

Step-by-Step Solution

1

Express the equation as a polynomial in standard form

Expanding

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

yields

x^4 - 6x^3 + 8x^2 + 2x - 1 = 0

.

Expanding allows the composite expression to be treated as a degree-4 polynomial.

2

Find a quadratic factor using fixed points of the iteration

Let

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

. The equation has the form

f(f(x)) = x

. Any solution to

f(x) = x

must satisfy this equation. Solving

x^2 - 3x + 1 = x

gives

x^2 - 4x + 1 = 0

.

If

f(x) = x

, then

f(f(x)) = f(x) = x

, so

x^2 - 4x + 1

must be a factor of the 4th-degree polynomial.

3

Factor the degree-4 polynomial completely

Polynomial division of

x^4 - 6x^3 + 8x^2 + 2x - 1

by

x^2 - 4x + 1

yields

(x^2 - 4x + 1)(x^2 - 2x - 1) = 0

.

Factoring reduces the 4th-degree equation into two quadratic equations.

4

Verify real roots and calculate their sum

For

x^2 - 4x + 1 = 0

, discriminant

D_1 = 12 > 0

and sum of roots is 4. For

x^2 - 2x - 1 = 0

, discriminant

D_2 = 8 > 0

and sum of roots is 2. The four roots are all real and distinct, giving a total sum of

4 + 2 = 6

.

Vieta's formulas provide the sum of roots for each quadratic factor without needing to compute individual radical expressions.

Key Concept

Polynomial Factoring and Composite Quadratic Equations

Estimated Time:2m 30s

Question 68Question →

If $x$ satisfies the equation $|8x + 13| = 10x + 5$ , what is the sum of all valid real solutions for $x$ ?

4

3

-1

-4

5

Show answer & explanation

Answer: 4

Answer

The sum of all valid real solutions for

x

is 4.

The solution

x = 4

is the only value of

x

that satisfies

|8x + 13| = 10x + 5

because evaluating both sides yields

45 = 45

. The alternative algebraic candidate,

x = -1

, produces a right-hand side of

-5

, which is impossible for an absolute value expression. Thus, the sum of all valid real solutions is simply 4.

Step-by-Step Solution

1

Split the absolute value equation into two linear case equations

Case 1:

8x + 13 = 10x + 5

; Case 2:

8x + 13 = -(10x + 5)

By definition of absolute value,

|u| = v

implies

u = v

or

u = -v

, provided

v \ge 0

.

2

Solve Case 1 for

x

2x = 8 \implies x = 4

Subtract

8x

and

5

from both sides to isolate

x

.

3

Solve Case 2 for

x

8x + 13 = -10x - 5 \implies 18x = -18 \implies x = -1

Expand the right-hand side and combine like terms to isolate

x

.

4

Test candidate solutions in the original equation to eliminate extraneous roots

For

x = 4

:

|8(4)+13| = |45| = 45

and

10(4)+5 = 45

(Valid). For

x = -1

:

|8(-1)+13| = |5| = 5

and

10(-1)+5 = -5

(Extraneous).

An absolute value output cannot equal a negative number.

5

Calculate the sum of all valid solutions

The only valid solution is

x = 4

, so the sum is

4

.

Only valid roots must be summed.

Key Concept

Absolute Value Equations and Checking for Extraneous Solutions

Question 69Question →

If $x$ is a real number satisfying the equation $x^2 + \frac{18}{x^2 + 3} = 8$ , what is the value of $x^2$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x^2

is 6.

Substituting

u = x^2

converts the equation to

u + \frac{18}{u+3} = 8

. Multiplying through by

(u+3)

yields

u^2 + 3u + 18 = 8u + 24

, which simplifies to the quadratic equation

u^2 - 5u - 6 = 0

. Factoring gives

(u-6)(u+1) = 0

, so

u = 6

or

u = -1

. Because

x

must be a real number,

x^2

must be non-negative, eliminating

u = -1

. Thus,

x^2 = 6

.

Step-by-Step Solution

1

Substitute a temporary variable

u = x^2

into the given equation

u + \frac{18}{u + 3} = 8

Using substitution simplifies the fourth-degree rational equation into a standard quadratic structure.

2

Clear the denominator by multiplying the entire equation by

(u + 3)

u(u + 3) + 18 = 8(u + 3)

, which expands to

u^2 + 3u + 18 = 8u + 24

Eliminating fractions allows rearranging terms into polynomial form.

3

Rearrange terms into standard quadratic form

au^2 + bu + c = 0

u^2 - 5u - 6 = 0

Moving all terms to one side prepares the expression for factoring.

4

Factor the quadratic polynomial into linear binomial factors

(u - 6)(u + 1) = 0

, giving

u = 6

or

u = -1

The roots of the transformed equation represent candidate values for

x^2

.

5

Apply the constraint that

x

is a real number to test candidate values

Since

x^2 \geq 0

for all real

x

,

u = -1

is rejected, leaving

u = x^2 = 6

Squaring any real number yields a non-negative result.

Key Concept

Solving rational equations reducible to quadratics via algebraic substitution and rejecting non-real roots.

Estimated Time:2m 0s

Question 70Question →

Two fulfillment centers, Center X and Center Y, process customer orders at constant individual rates of $x$ orders per hour and $y$ orders per hour, respectively, where $x > y > 0$ . Working together simultaneously, Center X and Center Y complete a total batch of $N$ orders in $T$ hours. In a separate shift, Center X works alone for $h$ hours (where $0 < h < T$ ) to process a portion of the $N$ orders, after which Center Y works alone for $k$ hours to finish all remaining orders in the batch.

Which of the following algebraic statements MUST be true? Select all that apply.

Select all that apply

The ratio of the processing rates satisfies

\frac{x}{y} = \frac{k - T}{T - h}

.

The processing rate of Center Y is expressed as

y = \frac{N(T - h)}{T(k - h)}

.

The total sequential processing time satisfies

h + k = 2T

.

The ratio of the processing rates simplifies directly to

\frac{x}{y} = \frac{k}{h}

.

The processing rate of Center X is expressed as

x = \frac{N(k - T)}{T(k + h)}

.

Show answer & explanation

Answer: The ratio of the processing rates satisfies $\frac{x}{y} = \frac{k - T}{T - h}$ .; The processing rate of Center Y is expressed as $y = \frac{N(T - h)}{T(k - h)}$ .

Answer

The valid algebraic statements are the rate ratio

\frac{x}{y} = \frac{k - T}{T - h}

and the expression for Center Y's rate

y = \frac{N(T - h)}{T(k - h)}

.

Equating the total work from combined operation

N = (x + y)T

and sequential operation

N = xh + yk

yields

x(T - h) = y(k - T)

, which proves that the ratio

\frac{x}{y} = \frac{k - T}{T - h}

must be true. Furthermore, substituting

x = \frac{N}{T} - y

into

xh + yk = N

gives

\frac{Nh}{T} + y(k - h) = N

, which simplifies to

y = \frac{N(T - h)}{T(k - h)}

. Therefore, both of these algebraic formulations are correct.

Step-by-Step Solution

1

Formulate the total work equations for both operational scenarios.

From simultaneous operation:

N = (x + y)T = xT + yT

. From sequential operation:

N = xh + yk

.

Total work completed in both scenarios equals the batch size

N

.

2

Equate the two expressions for

N

to derive the relationship between rates and times.

xT + yT = xh + yk \implies xT - xh = yk - yT \implies x(T - h) = y(k - T)

.

Grouping like rate terms enables solving for the rate ratio

\frac{x}{y}

.

3

Divide by

y(T - h)

to find the rate ratio

\frac{x}{y}

.

\frac{x}{y} = \frac{k - T}{T - h}.

Since

0 < h < T

,

T - h > 0

, so the division is valid and confirms the rate ratio relation.

4

Express

x

in terms of

y, N, T

and substitute into the sequential work equation.

x = \frac{N}{T} - y \implies \left(\frac{N}{T} - y\right)h + yk = N \implies \frac{Nh}{T} + y(k - h) = N

.

Eliminating

x

isolates

y

in terms of given parameters

N, T, h, k

.

5

Solve for

y

.

y(k - h) = N - \frac{Nh}{T} = N\left(1 - \frac{h}{T}\right) = \frac{N(T - h)}{T} \implies y = \frac{N(T - h)}{T(k - h)}

.

This establishes the exact formula for

y

.

Key Concept

Multi-variable system modeling of rate, time, and work constraints.

Estimated Time:2m 30s

Question 71Question →

How many integer values of $x$ satisfy the inequality $\frac{|2x - 7| - 3}{|x + 1| - 4} \leq 0$ ?

7

8

9

10

11

Show answer & explanation

Answer: 9

Answer

9 integer values satisfy the given inequality.

The quotient of two algebraic expressions is non-positive (

\leq 0

) when the numerator is zero (and denominator non-zero) or when the numerator and denominator have opposite signs. Testing these cases yields two disjoint intervals:

-5 < x \leq 2

and

3 < x \leq 5

. Within these intervals, the integers

-4, -3, -2, -1, 0, 1, 2, 4,

and

5

satisfy the inequality, yielding exactly 9 integer solutions.

Step-by-Step Solution

1

Determine where the numerator and denominator equal zero to set critical points and domain restrictions.

Numerator is zero at

|2x - 7| = 3 \implies x = 2

or

x = 5

. Denominator is zero at

|x + 1| = 4 \implies x = -5

or

x = 3

, which are excluded from the domain.

Division by zero is undefined, so

x \neq -5

and

x \neq 3

.

2

Analyze Case 1: Numerator

\geq 0

and Denominator

< 0

.

Numerator

\geq 0 \implies x \leq 2

or

x \geq 5

. Denominator

< 0 \implies -5 < x < 3

. Intersecting these gives

-5 < x \leq 2

.

A fraction is non-positive when the numerator and denominator have opposite signs.

3

Analyze Case 2: Numerator

\leq 0

and Denominator

> 0

.

Numerator

\leq 0 \implies 2 \leq x \leq 5

. Denominator

> 0 \implies x < -5

or

x > 3

. Intersecting these gives

3 < x \leq 5

.

A fraction is non-positive when the numerator and denominator have opposite signs.

4

Combine solution intervals and count integer solutions.

Combined solution set is

(-5, 2] \cup (3, 5]

. The integer values in this set are

-4, -3, -2, -1, 0, 1, 2, 4, 5

, giving a total count of 9.

Integrate all valid cases while excluding values that make the denominator zero.

Key Concept

Solving Rational Inequalities with Absolute Values

Estimated Time:2m 30s

Question 72Question →

An arithmetic sequence and a geometric sequence both have a first term equal to $2$ . The 3rd term of the arithmetic sequence is equal to the 2nd term of the geometric sequence, and the 7th term of the arithmetic sequence is equal to the 3rd term of the geometric sequence. If the common ratio of the geometric sequence is not equal to $1$ , what is the sum of the first $10$ terms of the arithmetic sequence?

20

30

54

65

130

Show answer & explanation

Answer: 65

Answer

The sum of the first 10 terms of the arithmetic sequence is 65.

By writing the equations for the given sequence terms (

2 + 2d = 2r

and

2 + 6d = 2r^2

), we substitute

d = r - 1

into the quadratic equation to find

r^2 - 3r + 2 = 0

. Since

r \neq 1

, we find

r = 2

, which gives a common difference

d = 1

. Applying the sum formula for an arithmetic sequence for

n = 10

yields

S_{10} = \frac{10}{2}(2(2) + 9(1)) = 5(13) = 65

.

Step-by-Step Solution

1

Express the 3rd and 7th terms of the arithmetic sequence and 2nd and 3rd terms of the geometric sequence.

Arithmetic terms:

a_3 = 2 + 2d

and

a_7 = 2 + 6d

. Geometric terms:

b_2 = 2r

and

b_3 = 2r^2

.

The

n

-th term of an arithmetic sequence is

a_n = a_1 + (n-1)d

and of a geometric sequence is

b_n = b_1 r^{n-1}

.

2

Set up the system of equations given by the problem.

2 + 2d = 2r \implies 1 + d = r \implies d = r - 1

, and

2 + 6d = 2r^2 \implies 1 + 3d = r^2

.

We are given

a_3 = b_2

and

a_7 = b_3

with

a_1 = b_1 = 2

.

3

Substitute

d = r - 1

into the second equation and solve for

r

.

1 + 3(r - 1) = r^2 \implies r^2 - 3r + 2 = 0 \implies (r - 1)(r - 2) = 0

. Since

r \neq 1

,

r = 2

.

Solving the quadratic equation yields two roots, but the problem excludes

r = 1

.

4

Determine the common difference

d

and compute the sum of the first 10 terms of the arithmetic sequence.

d = 2 - 1 = 1

.

S_{10} = \frac{10}{2}[2(2) + (10 - 1)(1)] = 5(4 + 9) = 65

.

The sum of the first

n

terms of an arithmetic sequence is given by

S_n = \frac{n}{2}[2a_1 + (n-1)d]

.

Key Concept

Combining Arithmetic and Geometric Sequence Formulas

Estimated Time:2m 0s

Question 73Question →

What is the sum of all distinct real solutions to the equation $(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$ ?

2

3

6

0

8

Show answer & explanation

Answer: 6

Answer

The sum of all distinct real solutions is 6.

By defining

u = x^2 - 3x

, the original equation reduces to

u^2 - 2u - 8 = 0

, which factors as

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

, giving

u = 4

or

u = -2

. Substituting back for

x

gives two quadratic equations:

x^2 - 3x - 4 = 0

(yielding roots

x = 4

and

x = -1

) and

x^2 - 3x + 2 = 0

(yielding roots

x = 1

and

x = 2

). The four distinct real roots are

-1, 1, 2,

and

4

, and their sum is

-1 + 1 + 2 + 4 = 6

.

Step-by-Step Solution

1

Use algebraic substitution to simplify the disguised quadratic equation.

Let

u = x^2 - 3x

. The equation becomes

u^2 - 2u - 8 = 0

.

Recognizing the repeating quadratic block

x^2 - 3x

transforms the 4th-degree polynomial into a standard 2nd-degree quadratic in terms of

u

.

2

Factor the quadratic equation in terms of

u

to find its roots.

(u - 4)(u + 2) = 0 \implies u = 4 \text{ or } u = -2

.

Finding the values of

u

establishes the two quadratic equations in terms of

x

that need to be solved.

3

Substitute

x^2 - 3x

back for

u

and solve for

x

in Case 1 (

u = 4

).

x^2 - 3x = 4 \implies x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0 \implies x = 4 \text{ or } x = -1

.

The discriminant

b^2 - 4ac = (-3)^2 - 4(1)(-4) = 25 > 0

, so there are two distinct real solutions:

x = 4

and

x = -1

.

4

Substitute

x^2 - 3x

back for

u

and solve for

x

in Case 2 (

u = -2

).

x^2 - 3x = -2 \implies x^2 - 3x + 2 = 0 \implies (x - 1)(x - 2) = 0 \implies x = 1 \text{ or } x = 2

.

The discriminant

b^2 - 4ac = (-3)^2 - 4(1)(2) = 1 > 0

, so there are two distinct real solutions:

x = 1

and

x = 2

.

5

Sum all distinct real solutions.

-1 + 1 + 2 + 4 = 6

.

All four solutions

\{-1, 1, 2, 4\}

are distinct and real.

Key Concept

Solving Disguised Quadratic Equations via Algebraic Substitution

Estimated Time:1m 30s

Question 74Question →

If the quadratic equation $3x^2 - kx + 48 = 0$ has equal real roots, where $k > 0$ , and the roots of the quadratic equation $x^2 + kx - m = 0$ differ by 10, what is the value of $m$ ?

Show answer & explanation

Answer: -119

Answer

The value of

m

is

-119

.

Setting the discriminant of

3x^2 - kx + 48 = 0

equal to zero gives

k^2 - 4(3)(48) = 0

, yielding

k = 24

since

k > 0

. Substituting

k = 24

into the second equation gives

x^2 + 24x - m = 0

. According to Vieta's formulas, the sum of roots is

r_1 + r_2 = -24

and the product of roots is

r_1 r_2 = -m

. Using the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

with the given root difference

|r_1 - r_2| = 10

yields

10^2 = (-24)^2 - 4(-m)

, which simplifies to

100 = 576 + 4m

. Subtracting 576 from both sides gives

4m = -476

, so

m = -119

.

Step-by-Step Solution

1

Find

k

using the discriminant condition for equal roots of

3x^2 - kx + 48 = 0

.

k = 24

A quadratic equation

ax^2 + bx + c = 0

has equal real roots when its discriminant

b^2 - 4ac = 0

. Here,

(-k)^2 - 4(3)(48) = 0

, so

k^2 = 576

. Since

k > 0

,

k = 24

.

2

Set up the algebraic relationship for the root difference of

x^2 + 24x - m = 0

.

(r_1 - r_2)^2 = 100

Given that the difference between the roots

r_1

and

r_2

is 10,

|r_1 - r_2| = 10

, which implies

(r_1 - r_2)^2 = 100

.

3

Apply Vieta's formulas to express

(r_1 - r_2)^2

in terms of

m

and solve.

m = -119

By Vieta's formulas for

x^2 + 24x - m = 0

, the sum of roots is

r_1 + r_2 = -24

and the product of roots is

r_1 r_2 = -m

. Using the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, substitute the known values:

100 = (-24)^2 - 4(-m) = 576 + 4m

. Solving

576 + 4m = 100

gives

4m = -476

, so

m = -119

.

Key Concept

Discriminant analysis and root difference identity via Vieta's formulas

Question 75Question →

For all real numbers $x > 1$ , the expression $\sqrt{x + 2\sqrt{x - 1}} - \sqrt{x - 2\sqrt{x - 1}}$ is equal to $2$ .

True

False

Show answer & explanation

Answer: False

Answer

The statement is False because the expression equals

2\sqrt{x-1}

(which is less than

2

) whenever

1 < x < 2

.

The statement asserts that the identity holds for all

x > 1

. However, evaluating

\sqrt{x - 2\sqrt{x-1}}

yields

|\sqrt{x-1} - 1|

. When

1 < x < 2

,

\sqrt{x-1} < 1

, so

|\sqrt{x-1} - 1| = 1 - \sqrt{x-1}

. Subtraction gives

2\sqrt{x-1} \neq 2

for all

x

in

(1, 2)

. Thus, the statement is false.

Step-by-Step Solution

1

Rewrite the expressions under the outer radicals as perfect squares.

x \pm 2\sqrt{x-1} = (x-1) \pm 2\sqrt{x-1} + 1 = (\sqrt{x-1} \pm 1)^2

Recognize the quadratic pattern

a^2 \pm 2ab + b^2

where

a = \sqrt{x-1}

and

b = 1

.

2

Apply the radical identity

\sqrt{u^2} = |u|

to remove the outer square roots.

\sqrt{x + 2\sqrt{x-1}} = |\sqrt{x-1} + 1| = \sqrt{x-1} + 1

and

\sqrt{x - 2\sqrt{x-1}} = |\sqrt{x-1} - 1|

The principal square root of a squared quantity is non-negative, requiring absolute value bars.

3

Analyze the absolute value

|\sqrt{x-1} - 1|

across the specified domain

x > 1

.

For

x \ge 2

,

\sqrt{x-1} \ge 1 \implies |\sqrt{x-1} - 1| = \sqrt{x-1} - 1

. For

1 < x < 2

,

\sqrt{x-1} < 1 \implies |\sqrt{x-1} - 1| = 1 - \sqrt{x-1}

.

The sign of the expression inside the absolute value changes at

x = 2

.

4

Evaluate the full expression for

1 < x < 2

.

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

Since

2\sqrt{x-1} < 2

when

1 < x < 2

, the statement does not hold for all real numbers

x > 1

.

Key Concept

Nested Radicals and the Principal Square Root Absolute Value Identity

\sqrt{u^2} = |u|

Question 76Question →

Consider the four values defined below based on arithmetic and geometric sequences. Arrange these four items in ascending order (from smallest numerical value to largest numerical value).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order from smallest to largest value is: (1) The 7th term of the geometric sequence (

192

), (2) The sum of the first 8 terms of the arithmetic sequence (

208

), (3) The sum of the infinite geometric series (

225

), and (4) The 50th term of the arithmetic sequence (

250

).

Evaluating each item yields: the 7th term of the geometric sequence equals 192, the sum of the first 8 terms of the arithmetic sequence equals 208, the sum of the infinite geometric series equals 225, and the 50th term of the arithmetic sequence equals 250. Placing these in ascending numerical order produces 192 < 208 < 225 < 250.

Step-by-Step Solution

1

Calculate the value of the 7th term of the geometric sequence

Using

g_7 = 3 \cdot 2^{7-1} = 3 \cdot 64 = 192

.

The

n\text{th}

term of a geometric sequence is given by

g_n = g_1 r^{n-1}

.

2

Calculate the sum of the first 8 terms of the arithmetic sequence

Using

S_8 = \frac{8}{2}[2(5) + (8-1)6] = 4[10 + 42] = 208

.

The sum of the first

n

terms of an arithmetic sequence is given by

S_n = \frac{n}{2}[2a_1 + (n-1)d]

.

3

Calculate the sum of the infinite geometric series

Using

S_\infty = \frac{150}{1 - 1/3} = \frac{150}{2/3} = 225

.

The sum of an infinite geometric series with

|r| < 1

is

S_\infty = \frac{a_1}{1-r}

.

4

Determine the common difference and the 50th term of the arithmetic sequence

Find

d = \frac{35 - 15}{7 - 3} = 5

, then

a_{50} = 15 + (50 - 3)5 = 15 + 235 = 250

.

Linear spacing between terms

a_k

and

a_m

yields

a_m - a_k = (m - k)d

.

5

Compare the calculated values to order them from smallest to largest

192 < 208 < 225 < 250

.

Arranging the quantities according to their numerical values gives the final sorted sequence.

Key Concept

Arithmetic and Geometric Sequences and Series

Question 77Question →

How many integer values of $x$ satisfy both $|2x - 3| < 9$ and $|x + 1| \ge 3$ ?

Show answer & explanation

Answer: 4

Answer

There are 4 integer values of

x

that satisfy both inequalities.

Solving

|2x - 3| < 9

gives

-3 < x < 6

, so the set of possible integer values is

\{-2, -1, 0, 1, 2, 3, 4, 5\}

. Solving

|x + 1| \ge 3

gives

x \ge 2

or

x \le -4

. Taking the intersection of these two conditions gives

x \in \{2, 3, 4, 5\}

, for a total of 4 integer values.

Step-by-Step Solution

1

Unfold the first absolute value inequality

|2x - 3| < 9

-9 < 2x - 3 < 9, which simplifies to -3 < x < 6

An absolute value inequality of the form |A| < B is equivalent to -B < A < B.

2

Unfold the second absolute value inequality

|x + 1| \ge 3

x + 1 \ge 3 or x + 1 \le -3, which simplifies to x \ge 2 or x \le -4

An absolute value inequality of the form |A| >= B is equivalent to A >= B or A <= -B.

3

List integer candidates and find the intersection

Candidates from first condition: {-2, -1, 0, 1, 2, 3, 4, 5}. Applying second condition (x >= 2 or x <= -4) leaves {2, 3, 4, 5}

The solution must satisfy both conditions simultaneously.

Key Concept

Solving systems of absolute value inequalities for integer solutions

Estimated Time:1m 30s

Question 78Question →

If $k$ is the product of all real solutions to the equation $|x^2 - 7| = 3x - 3$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The product of all real solutions is 8.

The equation

|x^2 - 7| = 3x - 3

requires that

3x - 3 \ge 0

, which gives the restriction

x \ge 1

. Solving the two cases

x^2 - 7 = 3x - 3

and

x^2 - 7 = -(3x - 3)

produces the candidate roots

x = 4, -1, 2, -5

. Eliminating the negative candidate roots leaves

x = 2

and

x = 4

as the only valid real solutions. Their product is

2 \times 4 = 8

.

Step-by-Step Solution

1

Determine the domain constraint for valid solutions.

Since the absolute value expression

|x^2 - 7|

cannot be negative,

3x - 3 \ge 0

, which requires

x \ge 1

.

An absolute value quantity

|A|

is always non-negative, so any equation of the form

|A| = B

requires

B \ge 0

for real solutions.

2

Solve the positive case

x^2 - 7 = 3x - 3

and test for extraneous solutions.

x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0

, giving

x = 4

(valid, as

4 \ge 1

) and

x = -1

(extraneous, as

-1 < 1

).

Solutions must satisfy the non-negativity constraint of the right-hand side.

3

Solve the negative case

x^2 - 7 = -(3x - 3)

and test for extraneous solutions.

x^2 + 3x - 10 = 0 \implies (x + 5)(x - 2) = 0

, giving

x = 2

(valid, as

2 \ge 1

) and

x = -5

(extraneous, as

-5 < 1

).

Solutions must satisfy the non-negativity constraint of the right-hand side.

4

Compute the product of the valid real solutions.

k = 2 \times 4 = 8

.

The question asks for the product of all valid real solutions.

Key Concept

Absolute Value Equations with Variable Expressions and Extraneous Solution Elimination

Question 79Question →

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

98

100

102

144

14

Show answer & explanation

Answer: 98

Answer

98

Rationalizing the denominator gives

x^{-1} = \sqrt{3} - \sqrt{2}

. Summing

x + x^{-1}

yields

2\sqrt{3}

. Squaring this sum gives

x^2 + 2 + x^{-2} = 12

, so

x^2 + x^{-2} = 10

. Squaring once more gives

x^4 + 2 + x^{-4} = 100

, which simplifies to

x^4 + x^{-4} = 98

.

Step-by-Step Solution

1

Find the reciprocal expression

x^{-1}

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

Rationalizing the denominator simplifies the expression for

x^{-1}

.

2

Calculate the sum

x + x^{-1}

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

Adding the original expression and its reciprocal eliminates the

\sqrt{2}

term.

3

Square

x + x^{-1}

to find

x^2 + x^{-2}

(x + x^{-1})^2 = x^2 + 2(x)(x^{-1}) + x^{-2} = x^2 + 2 + x^{-2} = (2\sqrt{3})^2 = 12 \implies x^2 + x^{-2} = 10

Using the algebraic identity

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

allows finding

x^2 + x^{-2}

.

4

Square

x^2 + x^{-2}

to find

x^4 + x^{-4}

(x^2 + x^{-2})^2 = x^4 + 2(x^2)(x^{-2}) + x^{-4} = x^4 + 2 + x^{-4} = 10^2 = 100 \implies x^4 + x^{-4} = 98

Squaring the second-power sum and subtracting the cross-term 2 yields the required value.

Key Concept

Rationalizing radical denominators and binomial squaring of reciprocals

Estimated Time:1m 30s

Question 80Question →

If real numbers $x$ and $y$ satisfy the system of linear equations:

7x + 4y = 113

3x + 6y = 87

what is the value of $x + y$ ?

Show answer & explanation

Answer: 20

Answer

The value of

x + y

is 20.

Adding the two equations yields

10x + 10y = 200

. Dividing the entire equation by 10 directly gives

x + y = 20

. Alternatively, solving the system yields

x = 11

and

y = 9

, whose sum is

11 + 9 = 20

.

Step-by-Step Solution

1

Add the two equations together to combine like terms.

(7x + 4y) + (3x + 6y) = 113 + 87 \implies 10x + 10y = 200

Recognizing that adding the equations creates symmetric coefficients of 10 for both

x

and

y

allows a direct solution for the sum

(x + y)

.

2

Divide the combined equation by 10.

\frac{10(x + y)}{10} = \frac{200}{10} \implies x + y = 20

Isolating

(x + y)

directly avoids the extra computation of solving for

x

and

y

individually.

Key Concept

Solving Systems of Linear Equations by Combination

Estimated Time:1m 30s

Algebra and Functions

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