Advanced Math Practice Questions — SAT — Page 6

Question 101Question →

For all $x > 3$ , which of the following expressions is equivalent to $\frac{x^2 - 9}{x^2 - 2x - 3} \cdot \frac{x + 1}{x}$ ?

\frac{x+3}{x+1}

\frac{x+3}{x}

\frac{x-3}{x}

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

Show answer & explanation

Answer: $\frac{x+3}{x}$

Answer

\frac{x+3}{x}

Factoring the numerator of the first fraction as

(x-3)(x+3)

and its denominator as

(x-3)(x+1)

allows the common factor

(x-3)

to cancel out, leaving

\frac{x+3}{x+1}

. Multiplying this result by the second fraction,

\frac{x+1}{x}

, permits the cancellation of the common factor

(x+1)

, which simplifies the entire expression to the equivalent form

\frac{x+3}{x}

.

Step-by-Step Solution

1

Factor the numerator and the denominator of the first rational expression:

\frac{x^2 - 9}{x^2 - 2x - 3}

.

The numerator factors as

(x - 3)(x + 3)

using the difference of squares identity. The denominator factors as

(x - 3)(x + 1)

by finding two numbers that multiply to

-3

and add to

-2

.

Factoring polynomials is necessary to identify and cancel common factors.

2

Simplify the first fraction by canceling the common factor

(x-3)

from both the numerator and denominator.

The first fraction simplifies to

\frac{x+3}{x+1}

.

Since

x > 3

, the term

x-3

is non-zero, making division by it valid.

3

Multiply the simplified first fraction by the second fraction:

\frac{x+3}{x+1} \cdot \frac{x+1}{x}

.

The common factor

(x+1)

in the numerator and denominator cancels out, resulting in

\frac{x+3}{x}

.

Canceling the common factor

(x+1)

yields the simplest equivalent expression.

Key Concept

Simplifying rational expressions by factoring polynomials and canceling common factors.

Question 102Question →

For all $x > 3$ , the expression $\frac{x^3 - 2x^2 - 9x + 18}{x^2 - 5x + 6} - \frac{x^3 + 3x^2 - 4x - 12}{x^2 + 5x + 6}$ is equivalent to the constant $k$ . What is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of the constant

k

is 5.

Factoring the numerators by grouping yields

x^3 - 2x^2 - 9x + 18 = (x-3)(x+3)(x-2)

and

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

. Dividing out their respective denominators

(x-2)(x-3)

and

(x+2)(x+3)

leaves the simplified linear expressions

x+3

and

x-2

. Subtracting these yields

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

.

Step-by-Step Solution

1

Factor the numerator and denominator of the first rational expression and cancel common factors.

x + 3

To simplify the first fraction by reducing it to its lowest terms.

2

Factor the numerator and denominator of the second rational expression and cancel common factors.

x - 2

To simplify the second fraction by reducing it to its lowest terms.

3

Subtract the second simplified expression from the first simplified expression.

5

To find the constant value equivalent to the given difference of rational expressions, ensuring to distribute the negative sign to all parts of the subtracted binomial.

Key Concept

Simplifying rational expressions by factoring cubic polynomials by grouping and quadratic trinomials

Question 103Question →

If the expression $\frac{6x^2 + 7x - 20}{2x - 3}$ is equivalent to $ax + b + \frac{c}{2x-3}$ for all $x \neq 1.5$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

Show answer & explanation

Answer: 15

Answer

The value of

a + b + c

is 15.

By dividing the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

, we find that the quotient is

3x + 8

and the remainder is

4

. Thus, the expression can be rewritten as

3x + 8 + \frac{4}{2x-3}

. Comparing this to the given expression

ax + b + \frac{c}{2x-3}

, we obtain

a = 3

,

b = 8

, and

c = 4

. Their sum is

3 + 8 + 4 = 15

.

Step-by-Step Solution

1

Set up the polynomial division of the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

.

Dividing

6x^2 + 7x - 20

by

2x - 3

.

To express the rational expression in terms of a polynomial quotient and a remainder.

2

Divide the first term of the numerator by the first term of the denominator to determine the first quotient term.

The first term is

3x

. Subtracting

3x(2x - 3)

from the numerator leaves

16x - 20

.

\frac{6x^2}{2x} = 3x

, and subtracting

6x^2 - 9x

from the polynomial leaves the next term to be divided.

3

Divide the leading term of the remaining expression by the leading term of the denominator to determine the constant term of the quotient.

The constant term is

8

. Subtracting

8(2x - 3)

from

16x - 20

leaves a remainder of

4

.

\frac{16x}{2x} = 8

, and subtracting

16x - 24

from

16x - 20

gives the final constant remainder.

4

Compare the quotient and remainder to the given form to identify

a

,

b

, and

c

.

a = 3

,

b = 8

, and

c = 4

.

The quotient is

3x + 8

and the remainder is

4

, which matches the form

ax + b + \frac{c}{2x-3}

.

5

Calculate the sum of

a

,

b

, and

c

.

3 + 8 + 4 = 15

.

The question asks for the value of

a + b + c

.

Key Concept

Polynomial division and rewrite of rational expressions

Question 104Question →

In the $xy$ -plane, the graphs of the linear function $3x + y = 6$ and the quadratic function $y = 3x^2 - 5x - 2$ intersect at two points. What is the sum of the $y$ -coordinates of these two intersection points?

10

\frac{2}{3}

14

2

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. By rewriting the linear equation as

y = 6 - 3x

and substituting it into the quadratic equation, we get

6 - 3x = 3x^2 - 5x - 2

. Collecting all terms on one side yields the quadratic equation

3x^2 - 2x - 8 = 0

. Factoring this equation gives

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

, which results in

x = 2

and

x = -\frac{4}{3}

. Substituting these

x

-values back into the linear equation gives the

y

-coordinates:

y = 6 - 3(2) = 0

and

y = 6 - 3(-\frac{4}{3}) = 10

. The sum of these

y

-coordinates is

0 + 10 = 10

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the linear equation.

y = 6 - 3x

Isolating

y

makes substitution into the quadratic equation straightforward.

2

Substitute the expression for

y

into the quadratic equation and set the equation to zero.

6 - 3x = 3x^2 - 5x - 2

, which simplifies to

3x^2 - 2x - 8 = 0

.

This forms a single quadratic equation in terms of

x

to find the

x

-coordinates of the intersection points.

3

Solve the quadratic equation by factoring.

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

, yielding

x = 2

and

x = -\frac{4}{3}

.

Factoring determines the values of

x

at the points of intersection.

4

Substitute the

x

-values back into the linear equation

y = 6 - 3x

to find the corresponding

y

-coordinates.

For

x = 2

:

y = 6 - 3(2) = 0

. For

x = -\frac{4}{3}

:

y = 6 - 3(-\frac{4}{3}) = 10

.

This identifies the coordinates of the two intersection points as

(2, 0)

and

(-\frac{4}{3}, 10)

.

5

Calculate the sum of the

y

-coordinates.

0 + 10 = 10

The question asks for the sum of the

y

-coordinates of the two intersection points.

Key Concept

Solving a system of linear and quadratic equations using substitution.

Estimated Time:1m 30s

Question 105Question →

A scientist studying a sample of a radioactive isotope determines that its mass decays exponentially. The mass of the isotope, in grams, is modeled by the function $M(t) = a \cdot b^t$ , where $t$ is the time, in days, since the study began, and $a$ and $b$ are constants. The table below shows the mass of the isotope for selected values of $t$ .

$t$ (days)	$M(t)$ (grams)
0	128
1	96
2	72

What is the value of $b$ ?

Show answer & explanation

Answer: 0.75

Answer

0.75

To find the constant

b

in the exponential decay function

M(t) = a \cdot b^t

, we can use the given table values. For

t = 0

,

M(0) = a \cdot b^0 = a = 128

. For

t = 1

,

M(1) = a \cdot b^1 = a \cdot b = 96

. Substituting

a = 128

into the second equation gives

128b = 96

. Dividing both sides by

128

yields

b = \frac{96}{128} = 0.75

(or

\frac{3}{4}

). We can verify this with

t = 2

:

M(2) = 128 \cdot (0.75)^2 = 128 \cdot 0.5625 = 72

, which matches the table.

Step-by-Step Solution

1

Determine the initial value

a

using the data point for

t = 0

.

a = 128

Substituting

t = 0

into

M(t) = a \cdot b^t

gives

M(0) = a \cdot b^0 = a

. Since the table shows

M(0) = 128

,

a

must equal

128

.

2

Use the data point for

t = 1

to write an equation for

b

.

128b = 96

Substituting

t = 1

and

a = 128

into the function gives

M(1) = 128 \cdot b^1 = 128b

. The table shows

M(1) = 96

.

3

Solve the equation to find

b

.

b = 0.75

Dividing both sides of

128b = 96

by

128

yields

b = \frac{96}{128}

, which simplifies to

0.75

(or

\frac{3}{4}

).

Key Concept

Finding the decay factor (base) of an exponential function from a table of values.

Question 106Question →

If $9^{2x + 1} = 27^{x - 3}$ , what is the value of $x$ ?

-5

-4

-11

10

Show answer & explanation

Answer: $-11$

Answer

-11

To solve

9^{2x + 1} = 27^{x - 3}

, express 9 and 27 with a common base of 3:

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

. Apply the power of a power rule to get

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

, which simplifies to

3^{4x + 2} = 3^{3x - 9}

. Equating the exponents gives

4x + 2 = 3x - 9

. Solving for

x

by subtracting

3x

and

2

from both sides yields

x = -11

. Therefore, the value of

x

is

-11

.

Step-by-Step Solution

1

Rewrite both sides of the equation with a common base.

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

Since 9 and 27 are both powers of 3 (

9 = 3^2

and

27 = 3^3

), expressing them with the same base allows the application of exponent rules.

2

Apply the power of a power rule

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

to simplify the exponents.

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

, which simplifies to

3^{4x+2} = 3^{3x-9}

Multiplying the inner exponent by the outer exponent simplifies the expression on both sides.

3

Set the exponents equal to each other.

4x + 2 = 3x - 9

Since the bases are equal and positive, the exponential expressions are equal if and only if their exponents are equal.

4

Solve the linear equation for

x

.

x = -11

Subtracting

3x

and

2

from both sides isolates the variable

x

.

Key Concept

Solving exponential equations by converting to a common base and equating exponents.

Question 107Question →

In the $xy$ -plane, the graphs of the linear equation $y = 3x + k$ , where $k$ is a constant, and the quadratic equation $y = x^2 - 2x + 7$ intersect at two points. If the distance between the two points of intersection is $\sqrt{130}$ , what is the value of $k$ ?

1

-4

4

10

Show answer & explanation

Answer: 4

Answer

4

The correct value of

k

is

4

. Equating the two equations yields the quadratic equation

x^2 - 5x + (7 - k) = 0

. Since the slope of the line is

3

, the difference in the y-coordinates of the intersection points is

3

times the difference in the x-coordinates. Applying the distance formula shows that the distance is equal to

\sqrt{10(x_2 - x_1)^2}

. Setting this equal to the given distance of

\sqrt{130}

yields

(x_2 - x_1)^2 = 13

. Substituting Vieta's relations (

x_1 + x_2 = 5

and

x_1 x_2 = 7 - k

) into the identity

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

results in

13 = 25 - 4(7 - k)

, which simplifies to

k = 4

.

Step-by-Step Solution

1

Equate the linear and quadratic equations to set up the intersection condition.

x^2 - 2x + 7 = 3x + k \dots \implies x^2 - 5x + (7 - k) = 0

The x-coordinates of the intersection points are the roots of this quadratic equation.

2

Express the distance between the two points

(x_1, y_1)

and

(x_2, y_2)

in terms of the difference of their x-coordinates.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(x_2 - x_1)^2 + (3(x_2 - x_1))^2} = \sqrt{10(x_2 - x_1)^2}

Since both points lie on the line

y = 3x + k

, the difference in their y-coordinates is

y_2 - y_1 = 3(x_2 - x_1)

.

3

Use the given distance to find the value of

(x_2 - x_1)^2

.

10(x_2 - x_1)^2 = 130 \implies (x_2 - x_1)^2 = 13

The distance between the intersection points is given as

\sqrt{130}

.

4

Apply Vieta's formulas and the algebraic identity for the squared difference of the roots to solve for

k

.

x_1 + x_2 = 5

and

x_1 x_2 = 7 - k

. Substituting these into

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

gives

13 = 5^2 - 4(7 - k) \implies 13 = 25 - 28 + 4k \implies 4k = 16 \implies k = 4

.

Vieta's formulas relate the coefficients of the quadratic equation to the sum and product of its roots.

Key Concept

Solving systems of linear and quadratic equations and finding constants based on geometric constraints.

Question 108Question →

If $x^2 - 9x + 18 = 0$ and $x > 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x

is 6.

Factoring the equation

x^2 - 9x + 18 = 0

gives

(x - 3)(x - 6) = 0

, which yields the solutions

x = 3

and

x = 6

. Applying the constraint

x > 4

, the only valid solution is 6.

Step-by-Step Solution

1

Factor the quadratic equation

x^2 - 9x + 18 = 0

.

(x - 3)(x - 6) = 0

Finding two numbers that multiply to 18 and add to -9 allows us to write the quadratic expression in its factored form.

2

Solve for the roots of the equation.

x = 3

or

x = 6

By the zero product property, setting each factor to zero yields the possible solutions for the equation.

3

Apply the given constraint

x > 4

.

x = 6

Since the question specifies that

x

must be greater than 4, the root

x = 3

is discarded, leaving

x = 6

as the only valid solution.

Key Concept

Solving quadratic equations by factoring and applying constraints.

Question 109Question →

The quadratic function $f$ is defined by $f(x) = (x - 5)(x - 11)$ . In the $xy$ -plane, the graph of $y = f(x)$ is a parabola with vertex $(h, k)$ . What is the value of $h$ ?

Show answer & explanation

Answer: 8

Answer

The value of

h

is

8

.

The given quadratic function is in factored form,

f(x) = (x - 5)(x - 11)

, which shows that the

x

-intercepts of the graph are at

x = 5

and

x = 11

. Because a parabola is symmetric, the

x

-coordinate of its vertex,

h

, is located exactly halfway between the two

x

-intercepts. Calculating the average of the

x

-intercepts gives

h = \frac{5 + 11}{2} = 8

.

Step-by-Step Solution

1

Find the

x

-intercepts of the parabola.

The

x

-intercepts are

x = 5

and

x = 11

.

The

x

-intercepts of a quadratic function in the form

f(x) = (x - r_1)(x - r_2)

occur at

x = r_1

and

x = r_2

because setting

f(x) = 0

yields these roots.

2

Calculate the

x

-coordinate of the vertex,

h

.

h = 8

Since a parabola is symmetric, its vertex lies on the axis of symmetry, which is the vertical line halfway between the

x

-intercepts. The average of

5

and

11

is

\frac{5 + 11}{2} = 8

.

Key Concept

Finding the vertex of a parabola from its factored form

Question 110Question →

A wildlife biologist models the population of a certain bird species in a nature reserve using the function $P(t) = 120(1.05)^t$ , where $P(t)$ represents the number of birds $t$ years after the study began. What does the number $120$ represent in this model?

The initial number of birds in the reserve when the study began

The number of birds added to the reserve each year

The percent by which the bird population increases each year

The number of years it takes for the bird population to double

Show answer & explanation

Answer: The initial number of birds in the reserve when the study began

Answer

The initial number of birds in the reserve when the study began

In the exponential function model

P(t) = a \cdot b^t

, the coefficient

a

is the value of the function at

t = 0

. Evaluating

P(0)

yields

120(1.05)^0 = 120

, which corresponds to the initial number of birds in the reserve when the study began.

Step-by-Step Solution

1

Identify the standard form of the exponential growth function

P(t) = a \cdot b^t

, where

a

represents the initial value and

b

represents the growth factor.

This helps map the given constants to their mathematical meanings.

2

Substitute

t = 0

to find the starting population

P(0) = 120(1.05)^0 = 120 \cdot 1 = 120

.

Finding the value at

t = 0

defines the initial state of the model.

3

Interpret the initial value in the context of the problem

The number

120

represents the number of birds present at the start of the study.

This links the mathematical result to the real-world scenario.

Key Concept

Interpreting components of an exponential function in context

Question 111Question →

In the $xy$ -plane, the vertex of the parabola $y = x^2 + bx + c$ is $V_1$ , and the vertex of the parabola $y = -x^2 + dx + e$ is $V_2$ , where $b$ , $c$ , $d$ , and $e$ are constants. The parabola $y = x^2 + bx + c$ passes through $V_2$ , and the parabola $y = -x^2 + dx + e$ passes through $V_1$ . If the distance between $V_1$ and $V_2$ is $2\sqrt{5}$ , and the line passing through $V_1$ and $V_2$ has a positive slope, what is the value of $b + d$ ?

2

4

8

16

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. By writing the equations of the parabolas in vertex form,

y = (x - h_1)^2 + k_1

and

y = -(x - h_2)^2 + k_2

, and using the fact that each parabola passes through the other's vertex, we establish that the vertical difference between the vertices is equal to the square of their horizontal difference:

k_2 - k_1 = (h_2 - h_1)^2

. Substituting this into the distance formula

(h_2 - h_1)^2 + (k_2 - k_1)^2 = 20

yields a quadratic equation in terms of

(h_2 - h_1)^2

, which solves to

(h_2 - h_1)^2 = 4

. Since the slope of the line segment connecting the vertices is positive, the horizontal difference

h_2 - h_1

must be positive, which gives

h_2 - h_1 = 2

. Finally, expressing the linear coefficients in terms of the vertex coordinates using the vertex formula gives

b = -2h_1

and

d = 2h_2

, so

b + d = 2(h_2 - h_1) = 2(2) = 4

.

Step-by-Step Solution

1

Write each parabola in vertex form using the vertices

V_1(h_1, k_1)

and

V_2(h_2, k_2)

.

f(x) = (x - h_1)^2 + k_1

and

g(x) = -(x - h_2)^2 + k_2

Expressing the functions in vertex form allows us to easily use the condition that each parabola passes through the other's vertex.

2

Substitute the vertex

V_2(h_2, k_2)

into

f(x)

and

V_1(h_1, k_1)

into

g(x)

.

k_2 = (h_2 - h_1)^2 + k_1

and

k_1 = -(h_1 - h_2)^2 + k_2

, both of which simplify to

k_2 - k_1 = (h_2 - h_1)^2

.

Since the points lie on the respective curves, their coordinates must satisfy the equations, establishing a relation between horizontal difference

\Delta h = h_2 - h_1

and vertical difference

\Delta k = k_2 - k_1

.

3

Use the distance formula for

V_1

and

V_2

with the given distance

2\sqrt{5}

and substitute

\Delta k = (\Delta h)^2

.

(\Delta h)^2 + ((\Delta h)^2)^2 = (2\sqrt{5})^2 \implies (\Delta h)^4 + (\Delta h)^2 - 20 = 0

. Solving this quadratic in terms of

(\Delta h)^2

yields

(\Delta h)^2 = 4

.

The distance formula connects the horizontal and vertical differences, which we can solve using our relation from the previous step.

4

Determine the horizontal difference

\Delta h

using the slope condition.

Since the slope

m = \frac{\Delta k}{\Delta h}

is positive and we know

\Delta k = 4 > 0

, we have

\Delta h = h_2 - h_1 = 2

.

The positive slope restriction determines that the horizontal translation is to the right from

V_1

to

V_2

.

5

Relate the standard form coefficients

b

and

d

to the vertex coordinates to find

b + d

.

For

f(x) = x^2 + bx + c

, the vertex x-coordinate is

h_1 = -\frac{b}{2} \implies b = -2h_1

. For

g(x) = -x^2 + dx + e

, the vertex x-coordinate is

h_2 = \frac{d}{2} \implies d = 2h_2

. Thus,

b + d = -2h_1 + 2h_2 = 2(h_2 - h_1) = 2(2) = 4

.

The vertex x-coordinate formula

h = -\frac{\text{coefficient of } x}{2 \times \text{coefficient of } x^2}

connects the standard form coefficients back to

h_1

and

h_2

.

Key Concept

Identifying vertex coordinates of quadratic functions and applying translations and the distance formula in the coordinate plane.

Question 112Question →

The quadratic equation $3x^2 + 12x - 15 = 0$ has solutions $r$ and $s$ , where $r > s$ . What is the value of $r - s$ ?

4

6

-4

-6

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. Dividing the given quadratic equation

3x^2 + 12x - 15 = 0

by 3 simplifies it to

x^2 + 4x - 5 = 0

. Factoring this expression gives

(x + 5)(x - 1) = 0

, which yields the solutions

x = 1

and

x = -5

. Since we are given that

r > s

, we define

r = 1

and

s = -5

. The difference between the roots is

r - s = 1 - (-5) = 1 + 5 = 6

.

Step-by-Step Solution

1

Divide the entire equation

3x^2 + 12x - 15 = 0

by 3 to simplify it.

x^2 + 4x - 5 = 0

Simplifying the equation makes it easier to factor by reducing the coefficients.

2

Factor the simplified quadratic equation

x^2 + 4x - 5 = 0

.

(x + 5)(x - 1) = 0

We look for two numbers that multiply to -5 and add to 4, which are 5 and -1.

3

Solve for the roots by setting each factor to zero, and assign the variables

r

and

s

such that

r > s

.

x = 1

and

x = -5

, meaning

r = 1

and

s = -5

.

Setting the factors to zero gives the solutions. Since the problem specifies r > s, the larger solution (1) is assigned to r and the smaller solution (-5) is assigned to s.

4

Calculate the value of

r - s

by substituting the values of

r

and

s

.

r - s = 1 - (-5) = 6

Subtracting s from r gives the final value of 6.

Key Concept

Solving quadratic equations by factoring and finding the difference between roots.

Alternative Method

Alternatively, Vieta's formulas can be used. For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

r + s = -b/a

and the product of the roots is

rs = c/a

. For

3x^2 + 12x - 15 = 0

, we find

r + s = -12/3 = -4

and

rs = -15/3 = -5

. The relationship between the sum, product, and difference of two numbers is given by the algebraic identity

(r - s)^2 = (r + s)^2 - 4rs

. Substituting our values gives

(r - s)^2 = (-4)^2 - 4(-5) = 16 + 20 = 36

. Since

r > s

, the difference

r - s

must be positive, so we take the positive square root:

r - s = \sqrt{36} = 6

.

Estimated Time:1m 0s

Question 113Question →

If $8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}$ , what is the value of $x$ ?

\frac{8}{3}

\frac{4}{3}

-\frac{4}{9}

5

Show answer & explanation

Answer: $\frac{8}{3}$

Answer

\frac{8}{3}

To solve the equation

8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}

, both bases can be written as powers of

2

because

8 = 2^3

and

\frac{1}{4} = 2^{-2}

. Substituting these values into the equation yields

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

. Applying the power rule of exponents,

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

, we multiply the exponents to get

2^{3x+6} = 2^{-2+6x}

. Since the bases are equal, we can set the exponents equal to each other:

3x + 6 = -2 + 6x

. Solving this linear equation by subtracting

3x

from both sides and adding

2

to both sides gives

8 = 3x

, which simplifies to

x = \frac{8}{3}

. Thus, the option with the value

\frac{8}{3}

is correct.

Step-by-Step Solution

1

Express both bases in the equation,

8

and

\frac{1}{4}

, as powers of

2

.

The base

8

is written as

2^3

and the base

\frac{1}{4}

is written as

2^{-2}

, yielding the equation

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

.

Expressing exponential terms with a common base is necessary to equate and solve their exponents.

2

Apply the power of a power exponent rule,

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

, to simplify both sides of the equation.

The equation becomes

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

, which simplifies to

2^{3x+6} = 2^{-2+6x}

.

This simplifies each side to a single base with a single exponent.

3

Since the bases are equal, set the exponents equal to each other and solve the resulting linear equation for

x

.

3x + 6 = -2 + 6x \implies 8 = 3x \implies x = \frac{8}{3}

.

Two exponential expressions with the same positive base are equal if and only if their exponents are equal.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Question 114Question →

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

The value of

k

is

12

.

The quadratic equation

x^2 - kx + 36 = 0

has exactly one real solution when its discriminant is equal to zero. The discriminant of

ax^2 + bx + c = 0

is

b^2 - 4ac

. For this equation,

a = 1

,

b = -k

, and

c = 36

. Setting the discriminant to zero gives

(-k)^2 - 4(1)(36) = 0

, which simplifies to

k^2 - 144 = 0

. Solving for

k

gives

k^2 = 144

, so

k = 12

or

k = -12

. Since the problem states that

k

is a positive constant, the correct value of

k

is

12

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

x^2 - kx + 36 = 0

.

a = 1

,

b = -k

, and

c = 36

.

To apply the discriminant formula, we need the values of

a

,

b

, and

c

.

2

Set the discriminant

\Delta = b^2 - 4ac

to

0

.

(-k)^2 - 4(1)(36) = 0

.

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Solve the equation for

k

.

k^2 - 144 = 0 \implies k^2 = 144 \implies k = 12

(since

k

is a positive constant).

Solving the equation yields

k = \pm 12

, and we select the positive value as specified in the problem.

Key Concept

Discriminant of a quadratic equation

Question 115Question →

A parabola in the $xy$ -plane has its vertex at $(3, 8)$ and passes through the point $(5, 16)$ . If the equation of the parabola is written in the form $y = a(x - 3)^2 + 8$ , what is the value of the constant $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Substituting the coordinates

(5, 16)

into

y = a(x - 3)^2 + 8

yields

16 = a(5 - 3)^2 + 8

. Simplifying the term

(5 - 3)^2

gives

4

, leading to the equation

16 = 4a + 8

. Subtracting 8 from both sides gives

8 = 4a

, and dividing by 4 gives the final answer of 2.

Step-by-Step Solution

1

Substitute the point

(5, 16)

into the equation.

16 = a(5 - 3)^2 + 8

Since the parabola passes through the point

(5, 16)

, these coordinates must satisfy the equation.

2

Simplify the squared term.

16 = 4a + 8

Subtract 3 from 5 to get 2, and square 2 to get 4.

3

Solve for the constant

a

.

a = 2

Subtract 8 from both sides to get

8 = 4a

, then divide by 4 to find

a

.

Key Concept

Determining a constant in a quadratic function given its vertex form and a point on the graph.

Question 116Question →

If $25^a = 125^b$ , where $a$ and $b$ are positive constants, what is the value of $\frac{a}{b}$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5

To find the value of

\frac{a}{b}

, we can rewrite the bases of the equation

25^a = 125^b

using a common base of 5:

(5^2)^a = (5^3)^b

. Applying the exponent power rule gives

5^{2a} = 5^{3b}

. Since the bases are identical, their exponents must be equal, so

2a = 3b

. Dividing both sides of this equation by

2b

yields the ratio

\frac{a}{b} = \frac{3}{2}

, which is

1.5

.

Step-by-Step Solution

1

Express the bases as powers of 5

(5^2)^a = (5^3)^b

To solve the equation, express both sides with a common base of 5 since

25 = 5^2

and

125 = 5^3

.

2

Apply the power rule of exponents

5^{2a} = 5^{3b}

According to the exponent rules,

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

.

3

Set the exponents equal to each other

2a = 3b

Because the bases on both sides of the equation are equal and positive, their exponents must also be equal.

4

Solve for the ratio

\frac{a}{b} = 1.5

Divide both sides of the equation

2a = 3b

by

2b

to isolate the ratio

\frac{a}{b}

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

Question 117Question →

A projectile is launched from a platform. The height $h(t)$ , in meters, of the projectile $t$ seconds after it is launched is modeled by the function $h(t) = -5(t - 2)^2 + 24$ . What is the maximum height, in meters, reached by the projectile?

2

24

-2

4

Show answer & explanation

Answer: 24

Answer

The maximum height reached by the projectile is 24 meters.

The given quadratic function is in vertex form,

h(t) = a(t - d)^2 + c

, where the vertex of the parabola is

(d, c)

. For the function

h(t) = -5(t - 2)^2 + 24

, the vertex is at

(2, 24)

. Because the leading coefficient

-5

is negative, the parabola opens downward, which means the vertical coordinate of the vertex represents the maximum value of the function. Therefore, the maximum height reached by the projectile is 24 meters.

Step-by-Step Solution

1

Identify the form of the quadratic function.

The function

h(t) = -5(t - 2)^2 + 24

is given in vertex form, which is

h(t) = a(t - d)^2 + c

.

Recognizing the vertex form allows direct identification of the vertex of the parabola.

2

Determine the vertex of the parabola.

For

h(t) = -5(t - 2)^2 + 24

, the vertex is

(2, 24)

.

The vertex coordinates

(d, c)

represent the input and output values at the turning point of the parabola.

3

Determine whether the vertex represents a maximum or a minimum.

Since the coefficient

a = -5

is negative, the parabola opens downward, meaning the vertex is the highest point on the graph.

The sign of the leading coefficient determines the concavity of the parabola.

4

Extract the maximum value of the function.

The maximum value is the vertical coordinate of the vertex, which is 24.

The height of the projectile is represented by the output of the function, which is maximized at the vertex.

Key Concept

Identifying the maximum value of a quadratic function from its vertex form.

Question 118Question →

In the quadratic equation $8x^2 - kx + 27 = 0$ , where $k$ is a positive constant, one of the roots is the square of the other root. What is the value of $k$ ?

Show answer & explanation

Answer: 30

Answer

30

For the quadratic equation

8x^2 - kx + 27 = 0

, Vieta's formulas state that the product of the roots is

\frac{27}{8}

and the sum of the roots is

\frac{k}{8}

. Letting the roots be

r

and

r^2

, we obtain the equation

r^3 = \frac{27}{8}

, which gives

r = \frac{3}{2}

. The other root is

r^2 = \frac{9}{4}

. Summing these roots gives

\frac{3}{2} + \frac{9}{4} = \frac{15}{4}

. Equating this sum to the formula for the sum of the roots yields

\frac{k}{8} = \frac{15}{4}

, which simplifies to

k = 30

.

Step-by-Step Solution

1

Determine the product of the roots using Vieta's formulas.

The product of the roots is

r_1 r_2 = \frac{27}{8}

.

For any quadratic equation

ax^2 + bx + c = 0

, the product of its roots is given by

\frac{c}{a}

.

2

Use the root relationship to solve for one of the roots.

r_2^3 = \frac{27}{8} \implies r_2 = \frac{3}{2}

.

Substituting

r_1 = r_2^2

into the product equation yields

r_2^3 = \frac{27}{8}

.

3

Determine the value of the second root.

r_1 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

.

The first root is the square of the second root.

4

Calculate the sum of the two roots.

r_1 + r_2 = \frac{9}{4} + \frac{3}{2} = \frac{15}{4}

.

The sum of the roots is needed to relate back to the linear coefficient of the quadratic equation.

5

Solve for the constant

k

using the sum of the roots.

\frac{k}{8} = \frac{15}{4} \implies k = 30

.

By Vieta's formulas, the sum of the roots is

-\frac{b}{a} = \frac{k}{8}

.

Key Concept

Quadratic Equations and Vieta's Formulas

Question 119Question →

In the quadratic equation $x^2 + px + q = 0$ , $p$ and $q$ are prime numbers. If the equation has two distinct integer roots, what is the value of $p + q$ ?

5

6

7

9

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. By Vieta's formulas, the sum of the roots is

-p

and the product of the roots is

q

. Since the roots are integers and

q

is prime, the roots must be

\{-1, -q\}

or

\{1, q\}

. The case of positive roots leads to

p + q = -1

, which is impossible for prime numbers. The case of negative roots leads to

p - q = 1

. The only prime numbers that differ by

1

are

3

and

2

. Therefore,

p = 3

and

q = 2

, and their sum is

5

.

Step-by-Step Solution

1

Apply Vieta's formulas to express the sum and product of the roots in terms of the coefficients.

For the equation

x^2 + px + q = 0

with distinct integer roots

r

and

s

, we have

r + s = -p

and

rs = q

.

Vieta's formulas state that for any quadratic equation

x^2 + bx + c = 0

with roots

r

and

s

,

r+s = -b

and

rs = c

.

2

Analyze the product equation

rs = q

using the properties of prime numbers.

Since

q

is prime, its only integer factors are

\pm 1

and

\pm q

. Thus, the integer roots

r

and

s

must be either

\{-1, -q\}

or

\{1, q\}

.

A prime number has no positive integer divisors other than 1 and itself.

3

Evaluate the first case where the roots are

1

and

q

.

If

r = 1

and

s = q

, then

r + s = 1 + q = -p

, which simplifies to

p + q = -1

. Since

p

and

q

are prime numbers, they must be positive (

p, q \ge 2

). Therefore,

p + q = -1

has no solution.

Prime numbers are positive integers greater than 1, so their sum cannot be negative.

4

Evaluate the second case where the roots are

-1

and

-q

.

If

r = -1

and

s = -q

, then

r + s = -1 - q = -p

, which simplifies to

p - q = 1

.

Substituting the negative roots into the sum equation yields a positive relationship between

p

and

q

.

5

Find the prime numbers

p

and

q

that satisfy

p - q = 1

.

The only consecutive prime numbers are

2

and

3

. Therefore,

q = 2

and

p = 3

. Both are prime numbers, and the equation

x^2 + 3x + 2 = 0

has distinct integer roots

-1

and

-2

.

Since all primes except 2 are odd, any two primes with a difference of 1 must include the only even prime, 2.

6

Calculate the sum of

p

and

q

.

The sum

p + q = 3 + 2 = 5

.

To answer the question, we add the two identified prime values.

Key Concept

Using Vieta's formulas and number theory properties of prime numbers to solve for coefficients of a quadratic equation.

Estimated Time:3m 0s

Question 120Question →

In the system of equations below, $k$ is a constant.

\begin{aligned} y &= 3x^2 - 12x + 7 \\ y &= kx - 5 \end{aligned}

If the system has two distinct real solutions, $(x_1, y_1)$ and $(x_2, y_2)$ , such that the average of the $x$ -coordinates of the solutions is $3$ , what is the value of $k$ ?

-6

-3

6

0

Show answer & explanation

Answer: 6

Answer

6

To find the value of

k

, we set the two equations equal to each other to solve for the

x

-coordinates of the intersection points:

3x^2 - 12x + 7 = kx - 5

. Moving all terms to one side gives the quadratic equation

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

. The sum of the roots

x_1 + x_2

is given by

-\frac{b}{a} = \frac{12+k}{3}

. The average of the roots is

\frac{x_1 + x_2}{2} = \frac{12+k}{6}

. Since the average of the

x

-coordinates is

3

, we set

\frac{12+k}{6} = 3

, which gives

12+k = 18

, and solving for

k

yields

6

. Evaluating the discriminant with

k = 6

gives

(-18)^2 - 4(3)(12) = 324 - 144 = 180 > 0

, confirming two distinct real solutions.

Step-by-Step Solution

1

Equate the two equations to find the

x

-coordinates of the intersection points.

3x^2 - 12x + 7 = kx - 5

The intersection points of the system occur where the

y

-values of the two functions are equal.

2

Rearrange the equation into standard quadratic form:

ax^2 + bx + c = 0

.

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

Grouping the like terms allows us to identify the quadratic coefficients

a = 3

,

b = -(12+k)

, and

c = 12

.

3

Express the sum and the average of the solutions in terms of

k

using Vieta's formulas.

Sum of solutions

x_1 + x_2 = -\frac{b}{a} = \frac{12+k}{3}

, so the average is

\frac{x_1 + x_2}{2} = \frac{12+k}{6}

.

Vieta's formulas state that the sum of the roots of a quadratic equation is

-\frac{b}{a}

. The average is half of the sum.

4

Set the average equal to

3

and solve for

k

.

\frac{12+k}{6} = 3 \implies 12+k = 18 \implies k = 6

.

The problem states that the average of the

x

-coordinates of the solutions is

3

.

5

Verify that

k = 6

yields two distinct real solutions by checking the discriminant.

For

k = 6

, the equation is

3x^2 - 18x + 12 = 0

. The discriminant is

D = (-18)^2 - 4(3)(12) = 324 - 144 = 180

. Since

D > 0

, two distinct real solutions exist.

A positive discriminant ensures that there are two distinct real solutions, satisfying the problem's condition.

Key Concept

Solving nonlinear systems of equations using quadratic properties and Vieta's formulas.

Advanced Math

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