Advanced Math Alıştırma Soruları — SAT — Sayfa 11

If $\frac{20}{x - 3} = 4$ , what is the value of $x + 2$ ?

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Cevap: 10

Cevap

10

To solve the equation

\frac{20}{x - 3} = 4

, multiply both sides by

x - 3

to get

20 = 4(x - 3)

. Distributing the 4 gives

20 = 4x - 12

. Adding 12 to both sides yields

32 = 4x

, and dividing by 4 gives

x = 8

. Substituting

x = 8

into the expression

x + 2

gives

8 + 2 = 10

.

Adım Adım Çözüm

1

Multiply both sides of the equation by the denominator

x - 3

.

20 = 4(x - 3)

To eliminate the fraction and rewrite the rational equation in linear form.

2

Distribute the constant on the right side of the equation.

20 = 4x - 12

To remove the parentheses by multiplying 4 by both

x

and

-3

.

3

Add 12 to both sides of the equation.

32 = 4x

To isolate the term with the variable

x

on one side of the equation.

4

Divide both sides of the equation by 4.

x = 8

To solve for the variable

x

.

5

Substitute the value of

x

into the expression

x + 2

.

10

The question asks for the value of the expression

x + 2

rather than just the variable

x

.

Anahtar Kavram

Solving rational equations by clearing the denominator and isolating the variable to evaluate algebraic expressions.

Soru 202Soru →

The population of a colony of bacteria is modeled by the function $P(t) = 250 \cdot 8^{\frac{t}{6} + 1}$ , where $P(t)$ is the estimated number of bacteria $t$ hours after the start of an experiment. The function can be rewritten in the equivalent form $P(t) = A \cdot 2^{\frac{m}{k}}$ , where $m$ is the number of 20-minute intervals that have elapsed since the start of the experiment, and $A$ and $k$ are constants. What is the value of $A + k$ ?

506

2,002

2,006

2,018

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Cevap: 2,006

Cevap

2,006

The correct answer is 2,006. By writing the base 8 as 2^3 and applying the exponent rules, the function becomes 250 * 2^(t/2 + 3) = 2000 * 2^(t/2). Since m represents 20-minute intervals and there are 3 intervals in an hour, we have t = m/3. Substituting this into the function yields 2000 * 2^(m/6), which gives A = 2000 and k = 6. The sum is 2000 + 6 = 2006.

Adım Adım Çözüm

1

Express the base 8 in terms of base 2 to match the target form.

P(t) = 250 \cdot (2^3)^{\frac{t}{6} + 1}

Converting to base 2 allows simplification of the exponent to match the required base of 2 in the target equation.

2

Apply the exponent power rule

(b^y)^z = b^{yz}

to simplify the expression.

P(t) = 250 \cdot 2^{3\left(\frac{t}{6} + 1\right)} = 250 \cdot 2^{\frac{t}{2} + 3}

Multiplying the exponents simplifies the expression into a single power of base 2.

3

Isolate the constant coefficient by separating the terms in the exponent using the rule

b^{x+y} = b^x \cdot b^y

.

P(t) = 250 \cdot 2^3 \cdot 2^{\frac{t}{2}} = 250 \cdot 8 \cdot 2^{\frac{t}{2}} = 2000 \cdot 2^{\frac{t}{2}}

This separates the initial coefficient multiplier from the variable growth term.

4

Convert the time variable from hours (

t

) to 20-minute intervals (

m

).

Since there are 3 twenty-minute intervals in one hour,

m = 3t

, which means

t = \frac{m}{3}

. Substituting this into the equation yields

P(t) = 2000 \cdot 2^{\frac{m/3}{2}} = 2000 \cdot 2^{\frac{m}{6}}

.

This expresses the population model in terms of the new time unit variable,

m

, as required.

5

Identify the constants

A

and

k

and calculate their sum.

A = 2000

,

k = 6

, so

A + k = 2000 + 6 = 2006

.

Comparing the final model to

A \cdot 2^{\frac{m}{k}}

directly gives the values of the constants.

Anahtar Kavram

Rewriting and interpreting exponential functions by manipulating bases and changing variables.

Tahmini Süre:2m 0s

Soru 203Soru →

In the system of quadratic equations below, $c$ is an integer constant:

2x^2 - 11x + c = 0

x^2 - 15x + 4c = 0

If the first equation has two distinct real solutions and the second equation has no real solutions, what is the value of $c$ ?

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Cevap: 15

Cevap

15

For the first equation

2x^2 - 11x + c = 0

to have two distinct real solutions, its discriminant must be positive:

\Delta_1 = (-11)^2 - 4(2)(c) = 121 - 8c > 0

, which gives

c < 15.125

. For the second equation

x^2 - 15x + 4c = 0

to have no real solutions, its discriminant must be negative:

\Delta_2 = (-15)^2 - 4(1)(4c) = 225 - 16c < 0

, which gives

c > 14.0625

. The only integer

c

that satisfies the inequality

14.0625 < c < 15.125

is

15

.

Adım Adım Çözüm

1

Set up the inequality for the first equation's discriminant to have two distinct real solutions.

c < 15.125

A quadratic equation

ax^2 + bx + c = 0

has two distinct real solutions if its discriminant,

b^2 - 4ac

, is greater than zero. For

2x^2 - 11x + c = 0

, this means

(-11)^2 - 4(2)(c) > 0

, simplifying to

121 - 8c > 0

, or

c < 15.125

.

2

Set up the inequality for the second equation's discriminant to have no real solutions.

c > 14.0625

A quadratic equation has no real solutions if its discriminant is less than zero. For

x^2 - 15x + 4c = 0

, the discriminant is

(-15)^2 - 4(1)(4c) < 0

, simplifying to

225 - 16c < 0

, or

c > 14.0625

.

3

Find the integer value of

c

that satisfies both inequalities.

c = 15

Combining the results from the previous steps, we get

14.0625 < c < 15.125

. Since

c

is specified to be an integer, the only integer that lies in this open interval is

15

.

Anahtar Kavram

Using the discriminant to determine the number of real solutions of quadratic equations

Soru 204Soru →

If $\sqrt{2x + 15} - 6 = x$ , what is the value of $x + 5$ ?

-3

-2

2

8

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Cevap: 2

Cevap

The correct answer is 2.

The correct answer is 2. To solve the equation

\sqrt{2x + 15} - 6 = x

, we first isolate the radical by adding

6

to both sides, obtaining

\sqrt{2x + 15} = x + 6

. Squaring both sides yields

2x + 15 = x^2 + 12x + 36

. Rearranging the terms to form a quadratic equation gives

x^2 + 10x + 21 = 0

, which factors as

(x + 7)(x + 3) = 0

. This gives two potential solutions:

x = -7

and

x = -3

. We must check both potential solutions in the original equation. For

x = -3

, the equation holds true:

\sqrt{2(-3) + 15} - 6 = 3 - 6 = -3

. For

x = -7

, it does not:

\sqrt{2(-7) + 15} - 6 = 1 - 6 = -5 \neq -7

. Thus,

x = -3

is the only valid solution. Finally, substituting this into the expression

x + 5

gives

-3 + 5 = 2

.

Adım Adım Çözüm

1

Isolate the radical term on one side of the equation.

\sqrt{2x + 15} = x + 6

Before squaring both sides, the radical term must be isolated to avoid cross-terms on that side.

2

Square both sides of the equation to eliminate the radical.

2x + 15 = (x + 6)^2 \implies 2x + 15 = x^2 + 12x + 36

Squaring a square root cancels the radical, allowing us to solve the resulting algebraic equation.

3

Rearrange the equation into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 + 10x + 21 = 0

Subtracting

2x

and

15

from both sides collects all terms on one side so the quadratic can be factored.

4

Factor the quadratic equation.

(x + 7)(x + 3) = 0 \implies x = -7 \text{ or } x = -3

Finding two numbers that multiply to 21 and add to 10 allows us to factor the quadratic and find potential solutions.

5

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

For

x = -3

,

\sqrt{2(-3) + 15} - 6 = -3

is true. For

x = -7

,

\sqrt{2(-7) + 15} - 6 = -7

is false (gives

-5 \neq -7

). Thus,

x = -3

is the only valid solution.

Squaring both sides can introduce extraneous solutions that do not satisfy the original equation.

6

Evaluate the expression

x + 5

using the valid solution.

-3 + 5 = 2

The question asks for the value of

x + 5

, not just the value of

x

.

Anahtar Kavram

Solving radical equations by isolating the radical, squaring both sides, checking for extraneous solutions, and evaluating algebraic expressions.

Soru 205Soru →

If $\frac{x}{x - 2} - \frac{8}{x + 2} = \frac{8}{x^2 - 4}$ , what is the value of $x$ ?

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Cevap: 4

Cevap

4

To solve the rational equation, multiply both sides by the least common denominator,

x^2 - 4

. This results in the equation

x(x + 2) - 8(x - 2) = 8

. Expanding and combining like terms yields the quadratic equation

x^2 - 6x + 8 = 0

. Factoring this equation gives

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

, which yields the potential solutions

x = 2

and

x = 4

. However,

x = 2

must be excluded because it makes the denominators of the original equation equal to zero. Therefore, the only valid solution is

x = 4

.

Adım Adım Çözüm

1

Determine the least common denominator (LCD) for the terms in the equation.

The LCD is

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

.

Multiplying the entire equation by the LCD clears all rational expressions.

2

Multiply the entire equation by the LCD to eliminate the denominators.

x(x + 2) - 8(x - 2) = 8

This simplifies the rational equation into a polynomial equation.

3

Expand, simplify, and solve the resulting quadratic equation.

x^2 - 6x + 8 = 0 \Rightarrow (x - 2)(x - 4) = 0 \Rightarrow x = 2 \text{ or } x = 4

Factoring the quadratic equation gives the potential solutions.

4

Substitute each potential solution into the original equation to check for extraneous solutions.

Substituting

x = 2

leads to division by zero, making the expression undefined. Substituting

x = 4

yields a valid identity:

\frac{4}{2} - \frac{8}{6} = \frac{2}{3}

. Therefore,

x = 4

is the only valid solution.

Values that make any denominator in the original equation equal to zero must be excluded.

Anahtar Kavram

Solving rational equations and checking for extraneous solutions.

Soru 206Soru →

In the quadratic equation $2x^2 - 11x + c = 0$ , $c$ is a constant. If the sum of the squares of the two real solutions to the equation is $\frac{85}{4}$ , what is the value of $c$ ?

9

18

-9

\frac{9}{2}

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Cevap: 9

Cevap

9

The sum of the solutions of the quadratic equation is

11/2

and the product of the solutions is

c/2

. The sum of the squares of the solutions is

(11/2)^2 - 2(c/2) = 121/4 - c

. Setting this equal to the given value of

85/4

and solving for

c

yields

c = 9

.

Adım Adım Çözüm

1

Express the sum and product of the solutions using Vieta's formulas.

For the quadratic equation

2x^2 - 11x + c = 0

, the sum of the solutions is

x_1 + x_2 = -\frac{b}{a} = \frac{11}{2}

, and the product of the solutions is

x_1 x_2 = \frac{c}{a} = \frac{c}{2}

.

Vieta's formulas relate the coefficients of a polynomial to the sum and product of its roots.

2

Use the algebraic identity for the sum of squares of two variables.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

This identity allows us to express the sum of squares in terms of the sum and product of the roots.

3

Substitute the expressions from Step 1 into the identity and set it equal to the given sum of squares.

\left(\frac{11}{2}\right)^2 - 2\left(\frac{c}{2}\right) = \frac{85}{4} \Rightarrow \frac{121}{4} - c = \frac{85}{4}

To solve for

c

using the given value of the sum of squares, which is

\frac{85}{4}

.

4

Isolate the constant

c

.

c = \frac{121}{4} - \frac{85}{4} = \frac{36}{4} = 9

To find the numerical value of

c

.

Anahtar Kavram

Vieta's Formulas and Root Identities

Soru 207Soru →

If the polynomial $p(x) = x^2 - kx + 12$ is divisible by $x - 3$ , where $k$ is a constant, what is the value of $k$ ?

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Cevap: 7

Cevap

7

According to the Factor Theorem, if a polynomial

p(x)

is divisible by

x - c

, then

p(c) = 0

. For the polynomial

p(x) = x^2 - kx + 12

to be divisible by

x - 3

, we must have

p(3) = 0

. Substituting

x = 3

into the polynomial gives

3^2 - k(3) + 12 = 0

. Simplifying this equation yields

9 - 3k + 12 = 0

, which combines to

21 - 3k = 0

. Solving for

k

gives

3k = 21

, which simplifies to

k = 7

.

Adım Adım Çözüm

1

Apply the Factor Theorem to find the root of the polynomial.

Since

x - 3

is a factor of

p(x)

, evaluating the polynomial at

x = 3

must equal 0, so

p(3) = 0

.

The Factor Theorem states that

x - c

is a factor of a polynomial

p(x)

if and only if

p(c) = 0

.

2

Substitute

x = 3

into

p(x) = x^2 - kx + 12

and set it equal to 0.

3^2 - k(3) + 12 = 0

This sets up an algebraic equation to solve for the unknown constant

k

.

3

Simplify the equation and solve for

k

.

9 - 3k + 12 = 0 \implies 21 - 3k = 0 \implies 3k = 21 \implies k = 7

Combine like terms and isolate

k

using basic algebraic operations.

Anahtar Kavram

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

.

Soru 208Soru →

The graph of the quadratic function $f$ in the $xy$ -plane is a parabola with vertex $(3, 12)$ . If the graph passes through the point $(5, 8)$ , what is the $y$ -value of the point on the graph where $x = 1$ ?

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Cevap: 8

Cevap

The correct answer is

8

.

The vertex of the parabola is given as

(3, 12)

, which means the axis of symmetry is the line

x = 3

. Since a parabola is symmetric with respect to its axis of symmetry, any two points on the parabola that are equidistant from this line must share the same

y

-coordinate. The given point has an

x

-coordinate of

5

, which is

5 - 3 = 2

units to the right of the axis of symmetry. The target point has an

x

-coordinate of

1

, which is

3 - 1 = 2

units to the left of the axis of symmetry. Because both points are exactly

2

units away from the axis of symmetry, their

y

-coordinates are equal. Therefore, the

y

-value of the point where

x = 1

is

8

.

Adım Adım Çözüm

1

Determine the axis of symmetry of the parabola.

The axis of symmetry is the vertical line

x = 3

.

The vertex of a parabola

(h, k)

always lies on its axis of symmetry, which is

x = h

.

2

Find the horizontal distance from the axis of symmetry to the given point.

The distance from

x = 3

to

x = 5

is

|5 - 3| = 2

units.

This measures how far the point is horizontally from the line of symmetry.

3

Find the horizontal distance from the axis of symmetry to the target point.

The distance from

x = 3

to

x = 1

is

|1 - 3| = 2

units.

This determines if the target point is symmetric to the given point.

4

Equate the y-values using symmetry.

The

y

-value at

x = 1

is

8

.

Because both

x = 5

and

x = 1

are

2

units away from the axis of symmetry, their corresponding

y

-values must be identical.

Anahtar Kavram

Symmetry of Quadratic Graphs

Soru 209Soru →

The function $f$ is defined by $f(x) = (x - 5)^2 + 3$ . If the graph of $y = f(x)$ is translated 4 units down in the $xy$ -plane to create the graph of the function $g$ , what is the vertex of the graph of $g$ ?

(1, 3)

(-5, -1)

(5, -1)

(5, 7)

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Cevap: $(5, -1)$

Cevap

The vertex of the graph of

g

is

(5, -1)

.

The vertex of a quadratic function of the form

f(x) = a(x - h)^2 + k

is

(h, k)

. For the function

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

, the vertex is

(5, 3)

. Translating the graph of a function vertically downward by 4 units decreases the

y

-coordinate of every point on the graph by 4, while the

x

-coordinate remains unchanged. Applying this transformation to the vertex coordinates yields

(5, 3 - 4)

, which simplifies to

(5, -1)

.

Adım Adım Çözüm

1

Identify the vertex of the original function

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

.

The vertex of the parabola

y = a(x - h)^2 + k

is

(h, k)

. For

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

, the vertex is

(5, 3)

.

Finding the original vertex is the starting point for applying the translation.

2

Apply the vertical translation of 4 units downward to the vertex coordinates.

A downward translation of 4 units subtracts 4 from the

y

-coordinate of the vertex:

(5, 3 - 4) = (5, -1)

.

Translating a graph vertically downward decreases the

y

-value of all its points by the translation distance.

Anahtar Kavram

Identifying the vertex of a quadratic function in vertex form and applying vertical translations.

Tahmini Süre:45s

Soru 210Soru →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has vertex $(3, 12)$ and passes through the point $(1, 0)$ . The function $g$ is defined by $g(x) = f(x + d) - 4$ , where $d$ is a constant. If the $y$ -intercept of the graph of $g$ is $(0, 5)$ and the vertex of the graph of $g$ lies in the second quadrant, what is the value of $d$ ?

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Cevap: 4

Cevap

The value of

d

is 4.

First, the equation of the function

f(x)

is determined in vertex form. Since the vertex of

f

is

(3, 12)

, we write

f(x) = a(x - 3)^2 + 12

. Substituting the point

(1, 0)

into this equation gives

0 = a(1 - 3)^2 + 12

, which yields

a = -3

. Thus,

f(x) = -3(x - 3)^2 + 12

. The transformation

g(x) = f(x + d) - 4

shifts the graph of

f

left by

d

units and down by 4 units, so the vertex of the graph of

g

is

(3 - d, 8)

. For this vertex to lie in the second quadrant, the

x

-coordinate must be negative, meaning

3 - d < 0

, or

d > 3

. The

y

-intercept of

g

is

(0, 5)

, so

g(0) = 5

. Since

g(0) = f(d) - 4

, we have

f(d) = 9

. Substituting

d

into

f(x)

gives

-3(d - 3)^2 + 12 = 9

, which simplifies to

(d - 3)^2 = 1

. Solving for

d

gives

d = 4

or

d = 2

. Since

d > 3

, the value of

d

must be 4.

Adım Adım Çözüm

1

Write the function

f(x)

in vertex form and substitute the point

(1, 0)

to solve for

a

.

f(x) = -3(x - 3)^2 + 12

The vertex

(h, k)

is given as

(3, 12)

, and the point

(1, 0)

lies on the graph.

2

Determine the vertex of

g(x)

based on the horizontal and vertical translations of

f(x)

.

The vertex of

g

is

(3 - d, 8)

.

The transformation

g(x) = f(x + d) - 4

shifts the vertex of

f(x)

left by

d

units and down by 4 units.

3

Establish the constraint on

d

using the quadrant of the vertex of

g

.

d > 3

For the vertex

(3 - d, 8)

to lie in the second quadrant, the

x

-coordinate must be negative.

4

Set up an equation for

d

using the

y

-intercept of

g(x)

.

f(d) = 9

The

y

-intercept is

(0, 5)

, so

g(0) = 5

. Substituting this into

g(x) = f(x + d) - 4

gives

5 = f(d) - 4

.

5

Solve

f(d) = 9

for

d

.

d = 2

or

d = 4

Substituting

d

into

f(x)

gives

-3(d - 3)^2 + 12 = 9

, which simplifies to

(d - 3)^2 = 1

.

6

Select the correct value of

d

that satisfies the quadrant constraint.

d = 4

Since

d > 3

, the value

d = 2

is discarded, leaving

d = 4

as the only valid solution.

Anahtar Kavram

Vertex form and transformations of quadratic functions

Soru 211Soru →

If $\sqrt{x + 2} = x$ , what is the value of $x - 1$ ?

-2

1

2

0

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of

x - 1

is

1

.

The valid solution to the radical equation is

x = 2

. Substituting this into the expression

x - 1

yields

2 - 1 = 1

.

Adım Adım Çözüm

1

Square both sides of the equation to eliminate the radical.

x + 2 = x^2

Squaring both sides removes the square root, allowing us to solve the resulting equation.

2

Rewrite the equation in standard quadratic form.

x^2 - x - 2 = 0

Moving all terms to one side sets the quadratic equation to zero, which is necessary for factoring.

3

Factor the quadratic equation.

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

, which gives the potential solutions

x = 2

and

x = -1

.

Factoring allows us to find the roots of the quadratic equation.

4

Check both potential solutions in the original equation to identify any extraneous solutions.

For

x = 2

,

\sqrt{2 + 2} = 2

is true. For

x = -1

,

\sqrt{-1 + 2} = -1 \implies 1 = -1

is false.

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical equation.

5

Calculate the value of

x - 1

using the valid solution.

2 - 1 = 1

The question asks for the value of the expression

x - 1

rather than just the variable

x

.

Anahtar Kavram

Radical Equations and Extraneous Solutions

Tahmini Süre:1m 0s

Soru 212Soru →

For all positive real numbers $x$ and $y$ , the expression $\frac{(4x^3 y^2)^2}{\sqrt{16x^8 y^6}}$ can be written in the equivalent form $ax^b y^c$ , where $a$ , $b$ , and $c$ are positive constants. What is the value of $a + b + c$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The expression simplifies to

4x^2 y^1

by applying the power of a product rule to the numerator to get

16x^6 y^4

, and simplifying the radical in the denominator to get

4x^4 y^3

. Dividing the terms yields

4x^2 y^1

. Identifying the coefficients and exponents gives

a = 4

,

b = 2

, and

c = 1

. The sum of these values is

4 + 2 + 1 = 7

.

Adım Adım Çözüm

1

Simplify the numerator of the expression.

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

Apply the power of a product rule

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

and the power of a power rule

(a^m)^n = a^{mn}

.

2

Simplify the denominator of the expression.

\sqrt{16x^8 y^6} = 4x^4 y^3

Take the square root of the coefficient and apply fractional exponent rules for positive variables:

\sqrt{16} = 4

,

\sqrt{x^8} = x^{8/2} = x^4

, and

\sqrt{y^6} = y^{6/2} = y^3

.

3

Divide the numerator by the denominator.

\frac{16x^6 y^4}{4x^4 y^3} = 4x^2 y^1

Divide the coefficients and subtract the exponents of the corresponding variables:

\frac{16}{4} = 4

,

x^{6-4} = x^2

, and

y^{4-3} = y^1

.

4

Identify the values of

a

,

b

, and

c

, and sum them.

a = 4

,

b = 2

,

c = 1

, and

a + b + c = 7

Match the simplified form

4x^2 y^1

to

ax^b y^c

to find

a

,

b

, and

c

, then calculate their sum.

Anahtar Kavram

Simplifying equivalent algebraic expressions using exponent rules and radical properties.

Tahmini Süre:1m 30s

Soru 213Soru →

If $x$ is a solution to the equation $\frac{2x - 3}{x - 1} - \frac{2}{x - 2} = \frac{1}{x^2 - 3x + 2}$ , what is the value of $2x - 1$ ?

1

5

6

9

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct answer is the value obtained by solving the rational equation for

x

and then calculating the expression

2x-1

. Multiplying the equation by the least common denominator

(x-1)(x-2)

yields the quadratic equation

2x^2 - 9x + 7 = 0

, which factors into

(2x - 7)(x - 1) = 0

. The potential solution

x = 1

is extraneous because it makes the denominators of the original equation equal to zero. Thus, the only valid solution is

x = 7/2

. Substituting this value into the expression gives

2(7/2) - 1 = 6

.

Adım Adım Çözüm

1

Identify the least common denominator (LCD) of the rational expressions in the equation.

The LCD is

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

. Since these denominators cannot be zero, we must have

x \neq 1

and

x \neq 2

.

Finding the LCD allows us to eliminate the denominators by multiplying both sides of the equation.

2

Multiply both sides of the equation by the LCD,

(x - 1)(x - 2)

, to clear the fractions.

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

This simplifies the rational equation into a polynomial equation.

3

Expand the products and simplify the equation.

(2x^2 - 7x + 6) - (2x - 2) = 1 \implies 2x^2 - 9x + 8 = 1

Expanding the terms is necessary to group like terms and form a quadratic equation.

4

Subtract 1 from both sides to write the quadratic equation in standard form.

2x^2 - 9x + 7 = 0

Standard form (

ax^2 + bx + c = 0

) is required to factor or apply the quadratic formula.

5

Factor the quadratic equation.

(2x - 7)(x - 1) = 0

, which gives potential solutions

x = \frac{7}{2}

and

x = 1

.

Factoring helps find the values of

x

that satisfy the polynomial equation.

6

Check the potential solutions against the original denominators to identify any extraneous solutions.

For

x = 1

, the denominators

x - 1

and

x^2 - 3x + 2

become zero, so

x = 1

is extraneous. For

x = \frac{7}{2}

, the denominators are non-zero, so

x = \frac{7}{2}

is the only valid solution.

Multiplying by variables can introduce extraneous solutions that make the original rational expressions undefined.

7

Calculate the value of the requested expression,

2x - 1

, using the valid solution

x = \frac{7}{2}

.

2\left(\frac{7}{2}\right) - 1 = 7 - 1 = 6

The question asks for the value of the expression

2x - 1

, not just the value of

x

.

Anahtar Kavram

Solving rational equations by clearing denominators, factoring the resulting quadratic equation, and checking for extraneous solutions.

Soru 214Soru →

If $\frac{x}{8} - \frac{2}{x} = 0$ and $x > 0$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct answer is 4. Adding the term

\frac{2}{x}

to both sides of the equation yields

\frac{x}{8} = \frac{2}{x}

. Cross-multiplying the terms gives

x^2 = 16

. Solving for

x

by taking the square root of both sides gives

x = 4

or

x = -4

. Since the question specifies that

x > 0

, the negative value is discarded, leaving 4 as the only valid solution.

Adım Adım Çözüm

1

Add

\frac{2}{x}

to both sides of the equation.

\frac{x}{8} = \frac{2}{x}

To isolate the rational terms on opposite sides of the equation.

2

Cross-multiply the terms.

x^2 = 16

To eliminate the denominators and form a quadratic equation.

3

Solve for

x

by taking the square root of both sides.

x = 4

or

x = -4

Taking the square root of 16 yields both positive and negative solutions.

4

Apply the constraint

x > 0

.

x = 4

The question specifies that

x

must be greater than 0, which excludes the negative solution.

Anahtar Kavram

Solving rational equations by isolating terms, cross-multiplying, and applying given constraints.

Soru 215Soru →

A parabola passes through the point $(0, 12)$ on the $y$ -axis and intersects the $x$ -axis at two distinct points, $P$ and $Q$ . The line connecting $P$ to the $y$ -intercept has a slope of $2$ , while the line connecting $Q$ to the $y$ -intercept has a slope of $-6$ . What is the maximum $y$ -value achieved by this parabola?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

The maximum

y

-value achieved by the parabola is 16.

The slopes of the lines connecting the

x

-intercepts

P(x_1, 0)

and

Q(x_2, 0)

to the

y

-intercept

C(0, 12)

determine their coordinates. The slope of the line through

P

and

C

is

\frac{12 - 0}{0 - x_1} = 2

, which gives

x_1 = -6

. The slope of the line through

Q

and

C

is

\frac{12 - 0}{0 - x_2} = -6

, which gives

x_2 = 2

. The factored form of the parabola is

y = a(x + 6)(x - 2)

. Using the

y

-intercept

C(0, 12)

, we find

12 = a(6)(-2)

, meaning

a = -1

. The

x

-coordinate of the vertex is the midpoint of the intercepts, which is

\frac{-6 + 2}{2} = -2

. Substituting

x = -2

into the equation gives

y = -(-2 + 6)(-2 - 2) = 16

. Since the leading coefficient is negative, this represents the maximum

y

-value.

Adım Adım Çözüm

1

Find the coordinates of

P(x_1, 0)

and

Q(x_2, 0)

using the slopes of the lines from these points to the

y

-intercept

C(0, 12)

.

P = (-6, 0)

and

Q = (2, 0)

The slope formula between

(x_1, 0)

and

(0, 12)

is

\frac{12 - 0}{0 - x_1} = 2

, which gives

x_1 = -6

. Similarly, for the second line,

\frac{12 - 0}{0 - x_2} = -6

, which gives

x_2 = 2

.

2

Write the quadratic equation in factored form using the

x

-intercepts.

y = a(x + 6)(x - 2)

Since the parabola intersects the

x

-axis at

(-6, 0)

and

(2, 0)

, these points are the roots of the quadratic function.

3

Solve for the leading coefficient

a

by substituting the

y

-intercept

(0, 12)

into the equation.

a = -1

Substituting

x = 0

and

y = 12

gives

12 = a(0 + 6)(0 - 2)

, which simplifies to

12 = -12a

, so

a = -1

.

4

Determine the

x

-coordinate of the vertex.

x = -2

The axis of symmetry of a parabola lies exactly midway between its

x

-intercepts:

x = \frac{-6 + 2}{2} = -2

.

5

Calculate the maximum

y

-value by evaluating the function at the

x

-coordinate of the vertex.

y = 16

Substituting

x = -2

into

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

gives

y = -(-2 + 6)(-2 - 2) = -(4)(-4) = 16

.

Anahtar Kavram

Using

x

-intercepts and coordinate geometry to find the vertex of a parabola.

Soru 216Soru →

A researcher models the population of a certain species of bacteria in a culture. The population $N(t)$ of bacteria $t$ hours after the culture is established is modeled by the function

N(t) = N_0 \cdot b^{\frac{t}{3}}

where $N_0$ is the initial population of the bacteria and $b$ is a constant. If the population of the bacteria increases by $44\%$ every 2 hours, what is the value of $b$ ?

1.200

1.275

1.660

1.728

Cevabı ve açıklamayı göster

Cevap: 1.728

Cevap

1.728

The correct answer is

1.728

. Since the population increases by

44\%

every 2 hours, the population at

t = 2

is

1.44

times the initial population, meaning

N(2) = 1.44 \cdot N_0

. Substituting

t = 2

into the model

N(t) = N_0 \cdot b^{\frac{t}{3}}

yields

N_0 \cdot b^{\frac{2}{3}} = 1.44 \cdot N_0

. Dividing both sides by

N_0

gives

b^{\frac{2}{3}} = 1.44

. Raising both sides to the power of

\frac{3}{2}

isolates

b

as

b = (1.44)^{\frac{3}{2}}

. Since

1.44 = (1.2)^2

, we can simplify this expression using exponent rules:

b = (1.2^2)^{\frac{3}{2}} = 1.2^3 = 1.728

.

Adım Adım Çözüm

1

Write the equation relating the population at time

t = 2

hours to the initial population at

t = 0

hours using the given percentage increase.

N(2) = 1.44 \cdot N_0

An increase of

44\%

means the population becomes

100\% + 44\% = 144\%

of its initial value, which corresponds to multiplying by a factor of

1.44

.

2

Substitute the function definition

N(t) = N_0 \cdot b^{\frac{t}{3}}

into the equation for

t = 2

.

N_0 \cdot b^{\frac{2}{3}} = 1.44 \cdot N_0

This allows us to set up an equation to solve for the constant

b

.

3

Divide both sides of the equation by

N_0

and isolate

b

.

b^{\frac{2}{3}} = 1.44

Since the initial population

N_0

is positive, we can divide both sides by

N_0

to isolate the exponential base term.

4

Solve for

b

by raising both sides of the equation to the power of

\frac{3}{2}

.

b = (1.44)^{\frac{3}{2}}

To solve for

b

, we multiply the exponent

\frac{2}{3}

by its reciprocal

\frac{3}{2}

.

5

Evaluate the expression

(1.44)^{\frac{3}{2}}

using exponent properties.

b = (1.2^2)^{\frac{3}{2}} = 1.2^3 = 1.728

Expressing

1.44

as

1.2^2

allows us to simplify the fractional exponent using the power of a power rule,

(x^a)^b = x^{ab}

.

Anahtar Kavram

Solving exponential equations by applying exponent rules and interpreting exponential growth factors in context.

Soru 217Soru →

The population of a certain species of plankton in a lake is modeled by the function

P(t) = P_0 \cdot (1.69)^{\frac{t}{d}}

where

P_0

is the initial population,

t

is the time, in days, since the population was first measured, and

d

is a positive constant. The population of the plankton increases by

30\%

every

h

hours, where

h

is a positive constant. If the model is accurate, what is the value of the ratio

\frac{h}{d}

?

0.5

2

12

48

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The correct value of the ratio is 12.

The correct answer shows the value of 12. Since

t

is measured in days, the time interval of

h

hours must be converted to

\frac{h}{24}

days. A

30\%

increase corresponds to a multiplier of

1.3

. Substituting

t = \frac{h}{24}

into the function gives

(1.69)^{\frac{h}{24d}} = 1.3

. Rewriting

1.69

as

1.3^2

yields

\left(1.3^2\right)^{\frac{h}{24d}} = 1.3

, which simplifies to

1.3^{\frac{h}{12d}} = 1.3^1

. Equating the exponents gives

\frac{h}{12d} = 1

, which solves to

\frac{h}{d} = 12

.

Adım Adım Çözüm

1

Express the growth interval in days to match the unit of

t

in the model.

Since there are 24 hours in a day, an interval of

h

hours is equivalent to

\frac{h}{24}

days.

The variable

t

in the model

P(t) = P_0 \cdot (1.69)^{\frac{t}{d}}

represents time in days, so all time variables must be in days.

2

Relate the percentage increase to the population multiplier.

An increase of

30\%

corresponds to a multiplier of

1 + 0.30 = 1.3

.

A

30\%

increase means the new population is

130\%

of the previous population.

3

Set up the equation using the given model at

t = \frac{h}{24}

.

P_0 \cdot (1.69)^{\frac{h}{24d}} = 1.3 \cdot P_0

, which simplifies to

(1.69)^{\frac{h}{24d}} = 1.3

.

At

t = \frac{h}{24}

days, the population must be

1.3

times the initial population

P_0

.

4

Rewrite the base of the exponential term and solve for the ratio.

Since

1.69 = 1.3^2

, the equation becomes

\left(1.3^2\right)^{\frac{h}{24d}} = 1.3^1

, which simplifies to

1.3^{\frac{2h}{24d}} = 1.3^1

, leading to

\frac{h}{12d} = 1

.

Expressing both sides with the same base allows for equating their exponents to solve for the target ratio.

5

Solve for the final ratio

\frac{h}{d}

.

\frac{h}{d} = 12

.

Multiplying both sides of the equation

\frac{h}{12d} = 1

by 12 isolates the ratio

\frac{h}{d}

.

Anahtar Kavram

Exponential Equations with Base Manipulation and Unit Conversion

Alternatif Yöntem

Instead of substituting a specific value for

t

, we can equate the daily growth factors. The model's daily growth factor is

(1.69)^{1/d}

. A

30\%

increase every

h

hours corresponds to a factor of

1.3

every

h

hours. Since there are

24

hours in a day, there are

\frac{24}{h}

such intervals in a day, yielding a daily growth factor of

(1.3)^{24/h}

. Setting the factors equal:

(1.69)^{1/d} = (1.3)^{24/h} \implies (1.3^2)^{1/d} = (1.3)^{24/h} \implies \frac{2}{d} = \frac{24}{h} \implies \frac{h}{d} = 12

.

Tahmini Süre:3m 0s

Soru 218Soru →

The value of a certain investment is modeled by the function $f(t) = 500(1.21)^{\frac{t}{2}}$ , where $f(t)$ represents the value of the investment, in dollars, $t$ years after the initial investment was made. Which of the following is the annual growth rate, as a percentage, of the investment?

21\%

10\%

10.5\%

46.41\%

Cevabı ve açıklamayı göster

Cevap: $10\%$

Cevap

The annual growth rate of the investment is

10\%

.

To find the annual growth rate, we need to rewrite the function

f(t) = 500(1.21)^{\frac{t}{2}}

in the form

f(t) = P(1 + r)^t

, where

P

is the initial value,

1 + r

is the annual growth factor, and

r

is the annual growth rate. Applying the power of a power rule for exponents, we can rewrite the expression as

f(t) = 500\left(1.21^{\frac{1}{2}}\right)^t

. Since raising a number to the power of

\frac{1}{2}

is equivalent to taking the square root, this simplifies to

f(t) = 500(\sqrt{1.21})^t = 500(1.1)^t

. The annual growth factor is

1.1

, which can be written as

1 + 0.1

. Therefore, the annual growth rate

r

is

0.1

, which corresponds to

10\%

.

Adım Adım Çözüm

1

Identify the given exponential function and rewrite the expression to isolate the annual growth factor.

The function is

f(t) = 500(1.21)^{\frac{t}{2}}

. Using exponent rules, we can rewrite the term

(1.21)^{\frac{t}{2}}

as

\left(1.21^{\frac{1}{2}}\right)^t

.

Isolating the variable

t

in the exponent allows us to identify the annual growth factor.

2

Calculate the base of the exponential term.

1.21^{\frac{1}{2}} = \sqrt{1.21} = 1.1

. Thus, the rewritten function is

f(t) = 500(1.1)^t

.

Simplifying the base gives us the actual growth factor per unit of time (per year).

3

Determine the percentage growth rate from the growth factor.

The growth factor is

1.1

, which is equal to

1 + 0.1

. The growth rate

r

is

0.1

. Converting

0.1

to a percentage gives

0.1 \times 100\% = 10\%

.

The growth rate is the decimal increase per year, which we convert to a percentage.

Anahtar Kavram

Manipulating exponential expressions to identify rates of growth or decay.

Soru 219Soru →

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

6x^{\frac{3}{4}} - 3

6x^2 - 3x

6x^2 - 3

5x^2 - 3

Cevabı ve açıklamayı göster

Cevap: $6x^2 - 3$

Cevap

6x^2 - 3

Distributing the term

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

to both terms inside the parentheses yields

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

. Multiplying the coefficients and adding the exponents according to the rule

x^a \cdot x^b = x^{a+b}

gives

6x^{\frac{1}{2} + \frac{3}{2}} - 3x^{\frac{1}{2} - \frac{1}{2}} = 6x^2 - 3x^0

. Since

x^0 = 1

for any positive

x

, the simplified equivalent expression is

6x^2 - 3

.

Adım Adım Çözüm

1

Distribute the term

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

to both terms inside the parentheses.

3x^{\frac{1}{2}} \cdot 2x^{\frac{3}{2}} - 3x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}}

Apply the distributive property

a(b - c) = ab - ac

to expand the expression.

2

Multiply the coefficients and apply the product rule for exponents,

x^a \cdot x^b = x^{a+b}

, to each product.

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

When multiplying terms with the same base, keep the base and add the exponents.

3

Simplify the arithmetic in the exponents and evaluate the resulting terms.

6x^2 - 3

Since

\frac{1}{2} + \frac{3}{2} = 2

and

\frac{1}{2} - \frac{1}{2} = 0

, the expression simplifies to

6x^2 - 3x^0

. Because

x > 0

,

x^0 = 1

, making the final expression

6x^2 - 3

.

Anahtar Kavram

Equivalent Algebraic Expressions

Alternatif Yöntem

Substitute a simple value for

x

, such as

x = 4

. The original expression evaluates to

3(4)^{1/2}(2(4)^{3/2} - 4^{-\frac{1}{2}}) = 3(2)(2(8) - \frac{1}{2}) = 6(16 - 0.5) = 6(15.5) = 93

. Evaluating the correct expression

6x^2 - 3

at

x = 4

yields

6(16) - 3 = 96 - 3 = 93

. Evaluating the other options at

x = 4

yields different values.

Tahmini Süre:1m 30s

Soru 220Soru →

In the quadratic equation $3x^2 + kx + 12 = 0$ , where $k$ is a constant, the difference between the two real solutions is $4$ . What is the value of $k^2$ ?

160

192

288

0

Cevabı ve açıklamayı göster

Cevap: 288

Cevap

The correct value of

k^2

is 288.

The correct value is 288. The difference between the two solutions

r_1

and

r_2

of the quadratic equation

ax^2 + bx + c = 0

is given by

|r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting

a = 3

,

b = k

, and

c = 12

, we get

|r_1 - r_2| = \frac{\sqrt{k^2 - 4(3)(12)}}{3} = \frac{\sqrt{k^2 - 144}}{3}

. Setting this equal to the given difference of

4

yields

\frac{\sqrt{k^2 - 144}}{3} = 4

. Multiplying by 3 gives

\sqrt{k^2 - 144} = 12

, and squaring both sides gives

k^2 - 144 = 144

, which simplifies to

k^2 = 288

. Alternatively, using Viete's formulas,

r_1 + r_2 = -\frac{k}{3}

and

r_1 r_2 = 4

. Using the identity

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

, we have

4^2 = (-\frac{k}{3})^2 - 4(4)

, which simplifies to

16 = \frac{k^2}{9} - 16

, leading to

\frac{k^2}{9} = 32

, or

k^2 = 288

.

Adım Adım Çözüm

1

Identify the coefficients of the quadratic equation

3x^2 + kx + 12 = 0

.

The coefficients are

a = 3

,

b = k

, and

c = 12

.

These values are needed to apply the difference of roots formula or Viete's relations.

2

State the formula for the difference between the two solutions

r_1

and

r_2

of a quadratic equation.

|r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting the coefficients gives

|r_1 - r_2| = \frac{\sqrt{k^2 - 4(3)(12)}}{3} = \frac{\sqrt{k^2 - 144}}{3}

.

This formula relates the difference of the roots directly to the coefficients of the equation.

3

Set the expression for the difference of the roots equal to the given difference of

4

and solve for

k^2

.

\frac{\sqrt{k^2 - 144}}{3} = 4 \Rightarrow \sqrt{k^2 - 144} = 12 \Rightarrow k^2 - 144 = 144 \Rightarrow k^2 = 288

.

Squaring both sides and isolating

k^2

yields its value.

Anahtar Kavram

Difference of roots and discriminant of a quadratic equation

Alternatif Yöntem

Use Viete's relations: The sum of the roots is

r_1 + r_2 = -\frac{k}{3}

and the product of the roots is

r_1 r_2 = \frac{12}{3} = 4

. The difference between the roots is given as

|r_1 - r_2| = 4

. Square this relation to get

(r_1 - r_2)^2 = 16

. Expand and rewrite the identity as

(r_1 + r_2)^2 - 4r_1 r_2 = 16

. Substituting the sum and product, we get

(-\frac{k}{3})^2 - 4(4) = 16 \Rightarrow \frac{k^2}{9} - 16 = 16 \Rightarrow \frac{k^2}{9} = 32 \Rightarrow k^2 = 288

.

Tahmini Süre:2m 0s

Advanced Math

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Alternatif Yöntem

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Sık Yapılan Hatalar

Alternatif Yöntem

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