Algebra and Functions Alıştırma Soruları — GMAT — Sayfa 4

Soru 61Soru →

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

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Custom Operator Evaluation and Order of Operations

Tahmini Süre:1m 15s

Soru 62Soru →

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

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Polynomial Factoring and the Zero-Product Property

Tahmini Süre:2m 0s

Soru 63Soru →

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

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Polynomial Factoring and Variable Cancellation Rules

Tahmini Süre:1m 30s

Soru 64Soru →

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

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving polynomial exponential equations of the form

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

by systematically testing base-exponent cases.

Tahmini Süre:2m 0s

Soru 65Soru →

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)

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

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

.

Adım Adım Çözüm

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)

.

Anahtar Kavram

Cyclic Functional Equations and System Substitution

Tahmini Süre:2m 30s

Soru 66Soru →

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

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Algebraic manipulation of quadratic equations using reciprocal square identities

Tahmini Süre:2m 0s

Soru 67Soru →

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

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

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Polynomial Factoring and Composite Quadratic Equations

Tahmini Süre:2m 30s

Soru 68Soru →

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

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Absolute Value Equations and Checking for Extraneous Solutions

Soru 69Soru →

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

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:2m 0s

Soru 70Soru →

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.

Geçerli olan tümünü seçin

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)}

.

Cevabı ve açıklamayı göster

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

Cevap

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.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Tahmini Süre:2m 30s

Soru 71Soru →

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

7

8

9

10

11

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving Rational Inequalities with Absolute Values

Tahmini Süre:2m 30s

Soru 72Soru →

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

Cevabı ve açıklamayı göster

Cevap: 65

Cevap

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

.

Adım Adım Çözüm

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]

.

Anahtar Kavram

Combining Arithmetic and Geometric Sequence Formulas

Tahmini Süre:2m 0s

Soru 73Soru →

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

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving Disguised Quadratic Equations via Algebraic Substitution

Tahmini Süre:1m 30s

Soru 74Soru →

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 $m$ ?

Cevabı ve açıklamayı göster

Cevap: -119

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Discriminant analysis and root difference identity via Vieta's formulas

Soru 75Soru →

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

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

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.

Adım Adım Çözüm

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

.

Anahtar Kavram

Nested Radicals and the Principal Square Root Absolute Value Identity

\sqrt{u^2} = |u|

Soru 76Soru →

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).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Arithmetic and Geometric Sequences and Series

Soru 77Soru →

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

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving systems of absolute value inequalities for integer solutions

Tahmini Süre:1m 30s

Soru 78Soru →

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

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Absolute Value Equations with Variable Expressions and Extraneous Solution Elimination

Soru 79Soru →

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

98

100

102

144

14

Cevabı ve açıklamayı göster

Cevap: 98

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Rationalizing radical denominators and binomial squaring of reciprocals

Tahmini Süre:1m 30s

Soru 80Soru →

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

7x + 4y = 113

3x + 6y = 87

what is the value of $x + y$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving Systems of Linear Equations by Combination

Tahmini Süre:1m 30s

Algebra and Functions

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