Advanced Math Practice Questions — SAT — Page 18

The table below shows some values for a cubic polynomial function $g$ .

$x$	$g(x)$
$-1$	$0$
$1$	$16$
$3$	$0$

In the $xy$ -plane, the graph of $y = g(x)$ is tangent to the $x$ -axis at $x = 3$ . What is the value of $g(0)$ ?

Show answer & explanation

Answer: 18

Answer

18

The correct value of

g(0)

is 18. By using the fact that

g(-1) = 0

, we establish

(x + 1)

as a factor. The tangency at

x = 3

tells us that

(x - 3)^2

is a factor. Writing the function as

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

and substituting

g(1) = 16

gives

8a = 16 \implies a = 2

. Evaluating

g(0)

yields

2(1)(9) = 18

.

Step-by-Step Solution

1

Determine the factors of the cubic polynomial

g(x)

using the given zeros and the tangency condition.

The factors are

(x + 1)

and

(x - 3)^2

, so the function is of the form

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

.

Since

g(-1) = 0

,

x = -1

is a root of the polynomial. The graph being tangent to the

x

-axis at

x = 3

indicates that

x = 3

is a root with a multiplicity of at least 2. Since

g(x)

is a cubic polynomial (degree 3), the multiplicity of the root at

x = 3

must be exactly 2.

2

Use the table value

g(1) = 16

to solve for the constant coefficient

a

.

a = 2

Substituting

x = 1

into

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

gives

g(1) = a(1 + 1)(1 - 3)^2 = 8a

. Setting this equal to the table value of 16 yields

8a = 16

, which simplifies to

a = 2

.

3

Evaluate the polynomial at

x = 0

using the fully determined function

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

.

18

To find

g(0)

, substitute

x = 0

into the expression:

g(0) = 2(0 + 1)(0 - 3)^2 = 2(1)(9) = 18

.

Key Concept

Identifying polynomial factors from graphs and tables, and analyzing root multiplicity (tangency vs. crossing).

Question 342Question →

The quadratic equation $x^2 - 12x + 4 = 0$ has solutions $x_1$ and $x_2$ . What is the value of $\frac{1}{x_1} + \frac{1}{x_2}$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By writing the expression

\frac{1}{x_1} + \frac{1}{x_2}

with a common denominator, we get

\frac{x_1 + x_2}{x_1 x_2}

. For the quadratic equation

x^2 - 12x + 4 = 0

, Vieta's formulas give the sum of the roots

x_1 + x_2 = 12

and the product of the roots

x_1 x_2 = 4

. Substituting these values into the fraction yields

\frac{12}{4} = 3

.

Step-by-Step Solution

1

Find a common denominator to combine the terms in the given expression.

\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2}

To express the target quantity in terms of the sum and product of the quadratic solutions.

2

Apply Vieta's formulas to find the sum and product of the solutions from the quadratic equation

x^2 - 12x + 4 = 0

.

x_1 + x_2 = 12

and

x_1 x_2 = 4

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

3

Substitute the sum and product values into the combined fraction.

\frac{12}{4} = 3

To compute the numerical value of the expression.

Key Concept

Sum and product of solutions of a quadratic equation

Question 343Question →

The graph of the quadratic function $f$ in the $xy$ -plane has $x$ -intercepts at $(-2, 0)$ and $(8, 0)$ . If the maximum value of $f(x)$ is $25$ , what is the value of $f(0)$ ?

Show answer & explanation

Answer: 16

Answer

16

The axis of symmetry of the quadratic function

f

lies halfway between the

x

-intercepts

x = -2

and

x = 8

, which is at

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

. Since the function has a maximum value of

25

, this maximum must occur at the vertex, giving the vertex coordinates

(3, 25)

. In vertex form, the function is

f(x) = a(x - 3)^2 + 25

. Substituting the

x

-intercept

(8, 0)

into the function yields

0 = a(8 - 3)^2 + 25

, which simplifies to

25a = -25

, or

a = -1

. Therefore, the equation of the function is

f(x) = -(x - 3)^2 + 25

. Evaluating this at

x = 0

gives

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

.

Step-by-Step Solution

1

Find the

x

-coordinate of the vertex (axis of symmetry)

x = 3

The axis of symmetry of a parabola is located exactly halfway between its

x

-intercepts:

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

.

2

Determine the vertex coordinates

(3, 25)

The maximum value of the quadratic function occurs at its vertex, so the

y

-coordinate of the vertex is the maximum value

25

.

3

Write the vertex form of the quadratic function

f(x) = a(x - 3)^2 + 25

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

4

Solve for the leading coefficient

a

a = -1

Substitute the

x

-intercept

(8, 0)

into the vertex form:

0 = a(8 - 3)^2 + 25 \implies 25a = -25 \implies a = -1

.

5

Find the value of

f(0)

f(0) = 16

Substitute

x = 0

into the function:

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

.

Key Concept

Using

x

-intercepts and the maximum value to determine the vertex and equation of a quadratic function.

Question 344Question →

If $x$ satisfies the equation below, what is the value of $x + 5$ ?

x - 4 = \sqrt{4x + 5}

Show answer & explanation

Answer: 16

Answer

The correct answer is 16.

Squaring both sides of the equation

x - 4 = \sqrt{4x + 5}

results in

(x - 4)^2 = 4x + 5

, which expands to

x^2 - 8x + 16 = 4x + 5

. Subtracting

4x + 5

from both sides gives the quadratic equation

x^2 - 12x + 11 = 0

. Factoring this quadratic equation yields

(x - 11)(x - 1) = 0

, giving candidate solutions of

x = 11

and

x = 1

. Checking these solutions in the original equation shows that

x = 11

is valid (

7 = 7

), whereas

x = 1

is extraneous (

-3 = 3

is false). Therefore, the only real solution is

x = 11

, and the value of the expression

x + 5

is

11 + 5 = 16

.

Step-by-Step Solution

1

Square both sides of the equation to remove the radical.

(x - 4)^2 = 4x + 5

Squaring a square root isolates the expression under the radical.

2

Expand the squared binomial on the left side.

x^2 - 8x + 16 = 4x + 5

Applying the distributive property to

(x - 4)(x - 4)

yields

x^2 - 8x + 16

.

3

Rearrange the equation to set it equal to zero.

x^2 - 12x + 11 = 0

Subtracting

4x + 5

from both sides simplifies the equation into standard quadratic form.

4

Factor the quadratic equation.

(x - 11)(x - 1) = 0

Finding two numbers that multiply to 11 and add to -12 gives -11 and -1.

5

Solve for the candidate values of x.

x = 11

and

x = 1

Setting each factor equal to zero yields the possible solutions.

6

Check the candidate values in the original equation to identify any extraneous solutions.

x = 11

is valid;

x = 1

is extraneous.

Substituting

x = 1

results in

-3 = 3

, which is false because the principal square root is always non-negative.

7

Evaluate the target expression using the valid solution.

11 + 5 = 16

The question asks for the value of

x + 5

, so we substitute the only valid solution,

x = 11

.

Key Concept

Solving radical equations by squaring both sides and checking for extraneous solutions.

Question 345Question →

If $9^{2x + 1} = \left(\frac{1}{27}\right)^{x - 2}$ , what is the value of $x$ ?

-8

1

\frac{4}{7}

-\frac{8}{7}

Show answer & explanation

Answer: $\frac{4}{7}$

Answer

\frac{4}{7}

The correct answer is

\frac{4}{7}

. By rewriting both sides of the equation with a common base of

3

, the equation becomes

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

. Applying the power of a power rule gives

3^{4x + 2} = 3^{-3x + 6}

. Since the bases are equal, the exponents must be equal, giving

4x + 2 = -3x + 6

. Solving for

x

results in

7x = 4

, which gives

x = \frac{4}{7}

.

Step-by-Step Solution

1

Rewrite each side of the equation with a common base of

3

.

9 = 3^2

and

\frac{1}{27} = 3^{-3}

, so the equation becomes

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

.

Before solving an exponential equation, it is helpful to express the bases in terms of their common prime base.

2

Apply the exponent power rule

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

to simplify the exponents.

3^{2(2x + 1)} = 3^{-3(x - 2)} \implies 3^{4x + 2} = 3^{-3x + 6}

.

Simplifying the expressions on both sides allows for equating the exponents directly.

3

Set the exponents equal to each other and solve the resulting linear equation.

4x + 2 = -3x + 6 \implies 7x = 4 \implies x = \frac{4}{7}

.

Since the bases are equal, the powers can only be equal if their exponents are equal.

Key Concept

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

Estimated Time:1m 30s

Question 346Question →

An object is launched from a platform. The function $h(t) = -5t^2 + 30t + 12$ models the height of the object, in meters, $t$ seconds after it was launched. What is the maximum height, in meters, reached by the object?

3

12

57

147

Show answer & explanation

Answer: 57

Answer

The maximum height reached by the object is 57 meters.

The given function

h(t) = -5t^2 + 30t + 12

represents a parabola opening downward because the coefficient of

t^2

is negative. The maximum value of this function occurs at its vertex. The time

t

at the vertex is determined using the formula

t = -\frac{b}{2a} = -\frac{30}{2(-5)} = 3

seconds. Substituting

t = 3

back into the function yields the maximum height:

h(3) = -5(3)^2 + 30(3) + 12 = -45 + 90 + 12 = 57

meters.

Step-by-Step Solution

1

Identify the coefficients of the quadratic function in standard form

h(t) = at^2 + bt + c

.

a = -5

,

b = 30

, and

c = 12

.

These coefficients are needed to calculate the vertex of the parabola.

2

Calculate the time

t

at which the maximum height occurs using the vertex formula

t = -\frac{b}{2a}

.

t = -\frac{30}{2(-5)} = 3

seconds.

Since the leading coefficient

a = -5

is negative, the parabola opens downward, meaning its vertex represents the maximum value.

3

Substitute

t = 3

back into the height function to find the maximum height.

h(3) = -5(3)^2 + 30(3) + 12 = -45 + 90 + 12 = 57

meters.

Evaluating the function at the time of the vertex gives the corresponding maximum height.

Key Concept

The maximum value of a quadratic function

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

(where

a < 0

) is the vertical coordinate of its vertex, which occurs at

x = -\frac{b}{2a}

.

Question 347Question →

The graph of the function $f$ in the $xy$ -plane has a vertex at $(-2, 7)$ . The function $g$ is defined by $g(x) = f(-x + 3) + 12$ . If the vertex of the graph of $y = g(x)$ is the point $(a, b)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 24

Answer

24

The vertex of the parent function

f(x)

is at

(-2, 7)

, which means

f(-2) = 7

. The transformed function is

g(x) = f(-x + 3) + 12

. The vertex of

g(x)

occurs when the input to

f

, which is

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

gives

-x = -5

, or

x = 5

, so the x-coordinate of the vertex of

g(x)

is

a = 5

. To find the y-coordinate

b

, we evaluate

g(5) = f(-(5) + 3) + 12 = f(-2) + 12 = 7 + 12 = 19

. Therefore, the vertex of the graph of

y = g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The value of

a + b

is

5 + 19 = 24

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of the transformed function

g(x)

by setting the inner expression equal to the x-coordinate of the original vertex.

a = 5

The vertex of

f(x)

is located at

x = -2

. For

g(x) = f(-x + 3) + 12

, the vertex occurs when the input to

f

,

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

yields

x = 5

.

2

Find the y-coordinate of the vertex of

g(x)

by evaluating

g(5)

.

b = 19

Substituting

x = 5

into the definition of

g(x)

gives

g(5) = f(-2) + 12

. Since the vertex of

f

is at

(-2, 7)

,

f(-2) = 7

. Thus,

g(5) = 7 + 12 = 19

.

3

Calculate the sum of the coordinates

a

and

b

.

24

The vertex of

g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The sum

a + b

is

5 + 19 = 24

.

Key Concept

Determining the coordinates of a transformed vertex using function notation.

Question 348Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are constants, has its vertex at $(4, 25)$ . If the positive $x$ -intercept of the graph of $f$ is $(d, 0)$ , what is the value of $d$ ?

Show answer & explanation

Answer: 9

Answer

The value of

d

is

9

.

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Since the vertex is

(4, 25)

and the coefficient of

x^2

is

-1

, the function is

f(x) = -(x-4)^2 + 25

. Setting

f(x) = 0

to find the

x

-intercepts yields

-(x-4)^2 + 25 = 0

, which simplifies to

(x-4)^2 = 25

. Taking the square root of both sides gives

x - 4 = 5

or

x - 4 = -5

. Solving these equations gives

x = 9

or

x = -1

. The positive

x

-intercept is

(9, 0)

, so the value of

d

is

9

.

Step-by-Step Solution

1

Write the function in vertex form.

f(x) = -(x-4)^2 + 25

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Here, the vertex is

(4, 25)

and the coefficient of

x^2

is

-1

, so

a = -1

,

h = 4

, and

k = 25

.

2

Set

f(x) = 0

to find the

x

-intercepts.

-(x-4)^2 + 25 = 0

The

x

-intercepts of a graph are the points where the function value is equal to 0.

3

Solve the equation for

x

.

x = 9

or

x = -1

Rearranging the equation gives

(x-4)^2 = 25

. Taking the square root of both sides gives

x-4 = 5

or

x-4 = -5

, which solves to

x = 9

or

x = -1

.

4

Identify the positive

x

-intercept coordinate

d

.

d = 9

The question asks for the positive

x

-intercept

(d, 0)

, which corresponds to

x = 9

.

Key Concept

Vertex form of a quadratic function and finding

x

-intercepts

Question 349Question →

The population of a species of fish in a lake can be modeled by the function $P(t) = P_0 \cdot 2^{\frac{t}{d}}$ , where $P_0$ is the initial population when the population was first measured, $t$ represents the time in years since it was first measured, and $d$ is a constant representing the doubling time in years. If the population of the fish doubles every 6 years, and the population after 18 years is 3,200, what was the initial population of the fish?

Show answer & explanation

Answer: 400

Answer

The initial population of the fish was 400.

By substituting the doubling time

d = 6

and the final population of 3,200 at

t = 18

into the exponential model

P(t) = P_0 \cdot 2^{\frac{t}{d}}

, we get

3,200 = P_0 \cdot 2^{\frac{18}{6}}

. Simplifying the exponent gives

3,200 = P_0 \cdot 2^3

, which simplifies further to

3,200 = 8P_0

. Dividing both sides by 8 yields

P_0 = 400

.

Step-by-Step Solution

1

Identify the values for the known variables from the word problem.

d = 6

years and at

t = 18

years,

P(18) = 3,200

.

To substitute these values into the exponential growth function model.

2

Substitute the known values into the exponential function

P(t) = P_0 \cdot 2^{\frac{t}{d}}

.

3,200 = P_0 \cdot 2^{\frac{18}{6}}

To set up an equation to solve for the unknown parameter

P_0

.

3

Simplify the exponent and calculate the growth factor.

3,200 = P_0 \cdot 2^3 \Rightarrow 3,200 = 8P_0

Reducing the fractional exponent simplifies the equation.

4

Solve for the initial population

P_0

by dividing both sides of the equation by 8.

P_0 = 400

Isolating

P_0

gives the initial population of the fish.

Key Concept

Using an exponential function to model real-world growth and solving for the initial value.

Question 350Question →

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

\frac{3}{7}

9

\frac{9}{7}

-\frac{9}{7}

Show answer & explanation

Answer: $\frac{3}{7}$

Answer

\frac{3}{7}

To solve the equation, we rewrite both bases using the common base

2

. Because

8 = 2^3

, the left side simplifies to

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

. Because

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

, the right side simplifies to

(2^{-2})^{2x - 3} = 2^{-4x + 6}

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

3x + 3 = -4x + 6

. Solving this linear equation by adding

4x

to both sides gives

7x + 3 = 6

. Subtracting

3

from both sides gives

7x = 3

. Dividing by

7

yields the value of

\frac{3}{7}

.

Step-by-Step Solution

1

Express both sides of the equation using a common base of

2

.

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

Since

8 = 2^3

and

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

, we can rewrite the terms with the same base to solve the exponential equation.

2

Apply the distributive property to simplify the exponents.

2^{3x + 3} = 2^{-4x + 6}

Multiplying the outer exponent by each term inside the parentheses simplifies the expression.

3

Set the exponents equal to each other and solve the linear equation for

x

.

3x + 3 = -4x + 6 \implies 7x = 3 \implies x = \frac{3}{7}

When bases are equal and positive (and not 1), their exponents must be equal.

Key Concept

Solving exponential equations by expressing both sides with a common base and equating exponents.

Question 351Question →

In the equation $(2x - 3)(x + 4) = k$ , $k$ is a constant. If $x = 2$ is a solution to the equation, what is the other solution to the equation?

-\frac{9}{2}

\frac{9}{2}

\frac{1}{2}

-9

Show answer & explanation

Answer: $-\frac{9}{2}$

Answer

-\frac{9}{2}

Substituting the known solution

x = 2

into the equation

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

yields

(2(2) - 3)(2 + 4) = k

, which simplifies to

(1)(6) = k

, so

k = 6

. Substituting this value back into the equation gives

(2x - 3)(x + 4) = 6

. Expanding the left side yields

2x^2 + 5x - 12 = 6

. Subtracting 6 from both sides places the quadratic equation in standard form:

2x^2 + 5x - 18 = 0

. Since

x = 2

is a root,

(x - 2)

is a factor. Factoring the quadratic gives

(x - 2)(2x + 9) = 0

. Setting the second factor equal to zero,

2x + 9 = 0

, yields the other solution,

x = -\frac{9}{2}

.

Step-by-Step Solution

1

Substitute the known solution

x = 2

into the equation to solve for

k

.

k = 6

Since

x = 2

is a solution, it must satisfy the equation

(2(2) - 3)(2 + 4) = k

, which simplifies to

(1)(6) = k

.

2

Substitute

k = 6

back into the original equation, expand the binomial product, and write the quadratic equation in standard form.

2x^2 + 5x - 18 = 0

Expanding the binomials gives

2x^2 + 5x - 12 = 6

. Subtracting 6 from both sides yields the standard form quadratic equation

2x^2 + 5x - 18 = 0

.

3

Factor the quadratic equation to find the other root.

(x - 2)(2x + 9) = 0

, yielding

x = 2

and

x = -\frac{9}{2}

Since

x = 2

is a solution,

(x - 2)

must be a factor. Dividing the quadratic by

(x - 2)

yields the other factor,

(2x + 9)

.

Key Concept

Solving quadratic equations by substituting a known root to determine constants, then rewriting and factoring the equation.

Alternative Method

Another way to find the other solution is to use the relationship between the coefficients of a quadratic equation and its roots. Once the equation is written in standard form as

2x^2 + 5x - 18 = 0

, the sum of the roots is given by

-\frac{b}{a} = -\frac{5}{2}

. Since one root is

2

, the other root

r

must satisfy

2 + r = -\frac{5}{2}

, which simplifies to

r = -\frac{5}{2} - 2 = -\frac{9}{2}

.

Estimated Time:1m 30s

Question 352Question →

An equation is shown below.

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

What is the real solution to the equation above?

Show answer & explanation

Answer: 2

Answer

The only real solution to the equation is

2

.

To solve the rational equation, we multiply all terms by the common denominator

x(x-1)

, assuming

x \neq 0

and

x \neq 1

. This results in the equation

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

. Expanding and simplifying this gives

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

. Factoring the quadratic expression yields

x(x-2) = 0

, which gives the potential solutions

x = 0

and

x = 2

. However, substituting

x = 0

back into the original equation causes a division by zero. Therefore,

x = 0

is an extraneous solution, and the only valid real solution is

2

.

Step-by-Step Solution

1

Find the common denominator for the terms in the rational equation.

The common denominator is

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

, which requires

x \neq 0

and

x \neq 1

.

Multiplying by the common denominator allows us to eliminate the fractions.

2

Multiply every term in the equation by the common denominator

x(x-1)

and simplify.

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

This clears the denominators and converts the rational equation into a polynomial equation.

3

Expand and simplify the resulting equation to standard quadratic form.

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

.

Grouping like terms is necessary to solve the quadratic equation.

4

Factor the quadratic equation to determine the potential solutions.

x(x-2) = 0

, giving potential solutions of

x = 0

and

x = 2

.

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

5

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

x = 0

is extraneous because it leads to division by zero. Thus, the only real solution is

x = 2

.

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

Key Concept

Solving rational equations and identifying extraneous solutions

Question 353Question →

For a real number $x$ , the equation $\sqrt{3x + 10} = x + 2$ is given. What is the value of the expression $x - 1$ ?

2

1

-4

-3

Show answer & explanation

Answer: 1

Answer

The correct answer is the value 1, which is obtained by evaluating the expression for the only valid solution to the equation.

To solve the given radical equation, square both sides to obtain a quadratic equation, which simplifies to

x^2 + x - 6 = 0

. Factoring this equation yields potential solutions of

x = 2

and

x = -3

. Substituting

x = 2

back into the original equation produces a true statement, confirming it as a valid solution. Substituting

x = -3

results in an inequality, identifying it as an extraneous solution. Evaluating the expression for the valid solution yields

2 - 1 = 1

.

Step-by-Step Solution

1

Eliminate the radical by squaring both sides of the equation.

3x + 10 = (x + 2)^2

, which expands to

3x + 10 = x^2 + 4x + 4

.

Squaring both sides removes the square root, allowing the equation to be solved algebraically.

2

Move all terms to one side to set the quadratic equation to zero.

x^2 + x - 6 = 0

.

A quadratic equation must be in the form

ax^2 + bx + c = 0

to solve it by factoring.

3

Factor the quadratic equation.

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

, giving potential solutions of

x = -3

and

x = 2

.

Factoring allows us to find the roots that satisfy the quadratic relationship.

4

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

For

x = 2

,

\sqrt{3(2) + 10} = 2 + 2 \Rightarrow 4 = 4

(valid). For

x = -3

,

\sqrt{3(-3) + 10} = -3 + 2 \Rightarrow 1 = -1

(invalid). Thus,

x = -3

is extraneous and

x = 2

is the only valid solution.

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

5

Evaluate the requested expression using the valid solution.

x - 1 = 2 - 1 = 1

.

The question asks for the value of the expression rather than the value of the variable itself.

Key Concept

Radical and Rational Equations

Question 354Question →

For a third-degree polynomial $p(x)$ , the expression $x^2 - 4x + 4$ is a factor. When $p(x)$ is divided by $x - 3$ , the remainder is $5$ , and when $p(x)$ is divided by $x + 1$ , the remainder is $-27$ . What is the remainder when $p(x)$ is divided by $x - 1$ ?

Show answer & explanation

Answer: 1

Answer

The remainder when the polynomial is divided by

x - 1

is

1

.

The correct answer is

1

. The polynomial is expressed as

p(x) = (x-2)^2(ax + b)

since it is a third-degree polynomial with a factor of

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

. Applying the Remainder Theorem, we evaluate the polynomial at

x = 3

and

x = -1

, giving the system of equations

3a + b = 5

and

-a + b = -3

. Solving this system yields

a = 2

and

b = -1

, meaning the polynomial is

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

. Finally, the remainder when

p(x)

is divided by

x - 1

is

p(1) = (1-2)^2(2(1) - 1) = 1

.

Step-by-Step Solution

1

Express the third-degree polynomial in terms of its known quadratic factor.

p(x) = (x-2)^2(ax + b)

Since the polynomial is of degree 3 and has a quadratic factor of

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

, the remaining factor must be linear, of the form

ax+b

.

2

Apply the Remainder Theorem to set up the system of equations.

p(3) = 5

and

p(-1) = -27

The Remainder Theorem states that the remainder of a polynomial

p(x)

when divided by

x - c

is equal to

p(c)

.

3

Substitute the values of

x

into the polynomial expression.

3a + b = 5

and

9(-a + b) = -27

Evaluating

p(3) = (3-2)^2(3a + b) = 5

yields

3a + b = 5

, and evaluating

p(-1) = (-1-2)^2(-a + b) = -27

yields

9(-a + b) = -27

.

4

Simplify the second equation and solve the system of linear equations.

a = 2

and

b = -1

Dividing the second equation by

9

gives

-a + b = -3

. Subtracting this from the first equation (

3a + b = 5

) gives

4a = 8

, which means

a = 2

. Substituting

a = 2

into the first equation yields

b = -1

.

5

Calculate the remainder when

p(x)

is divided by

x - 1

.

p(1) = 1

According to the Remainder Theorem, the remainder of

p(x)

divided by

x - 1

is

p(1)

. Substituting

x = 1

into

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

yields

(1-2)^2(2(1) - 1) = 1

.

Key Concept

Using the Remainder Theorem and known factors of a polynomial to solve for unknown coefficients.

Question 355Question →

In the quadratic equation $x^2 - 10x + c = 0$ , $c$ is a constant. If the two real solutions to the equation have a difference of 6, what is the value of $c$ ?

Show answer & explanation

Answer: 16

Answer

16

The correct answer is 16. By using the quadratic formula, the two solutions of the equation

x^2 - 10x + c = 0

are

5 + \sqrt{25 - c}

and

5 - \sqrt{25 - c}

. The difference between these two solutions is

2\sqrt{25 - c}

. Given that the difference is 6, we set

2\sqrt{25 - c} = 6

, which simplifies to

\sqrt{25 - c} = 3

. Squaring both sides gives

25 - c = 9

, which yields

c = 16

.

Step-by-Step Solution

1

Use the quadratic formula to express the solutions of

x^2 - 10x + c = 0

.

The solutions are

x = 5 \pm \sqrt{25 - c}

.

This expresses the roots of the quadratic equation in terms of the constant

c

.

2

Set the difference between the two solutions equal to 6.

(5 + \sqrt{25 - c}) - (5 - \sqrt{25 - c}) = 6

, which simplifies to

2\sqrt{25 - c} = 6

.

We are given that the two real solutions have a difference of 6.

3

Solve the equation

2\sqrt{25 - c} = 6

for

c

.

\sqrt{25 - c} = 3 \implies 25 - c = 9 \implies c = 16

.

This isolates the constant

c

using standard algebraic operations.

Key Concept

Solving quadratic equations and using properties of roots.

Alternative Method

Alternatively, we can use the relationship between the roots of a quadratic equation. If the roots are

x_1

and

x_2

, then

x_1 + x_2 = 10

and

x_1 x_2 = c

. Using the identity

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

, we substitute the given values:

(6)^2 = (10)^2 - 4c

. This simplifies to

36 = 100 - 4c

, which gives

4c = 64

, or

c = 16

.

Estimated Time:1m 30s

Question 356Question →

An equation is shown below.

\frac{x}{x-3} - \frac{2}{x+1} = \frac{8}{x^2-2x-3}

What is the real solution to the equation above?

Show answer & explanation

Answer: 2

Answer

The only real solution is 2.

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

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

. This yields

x(x+1) - 2(x-3) = 8

. Simplifying this equation gives

x^2 - x - 2 = 0

. Factoring the quadratic expression gives

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

, which yields potential solutions of

x = 2

and

x = -1

. However, substituting

x = -1

into the original equation results in division by zero, making it an extraneous solution. Substituting

x = 2

into the original equation is valid, so the only real solution is 2.

Step-by-Step Solution

1

Factor the quadratic denominator on the right side of the equation.

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

This helps identify the least common denominator of the rational terms.

2

Multiply all terms of the equation by the least common denominator,

(x - 3)(x + 1)

, to eliminate the denominators.

x(x + 1) - 2(x - 3) = 8

Multiplying by the LCD clears the rational expressions, converting the equation into a polynomial equation, under the restriction that

x \neq 3

and

x \neq -1

.

3

Expand and simplify the resulting equation.

x^2 + x - 2x + 6 = 8 \Rightarrow x^2 - x - 2 = 0

This puts the equation into standard quadratic form:

ax^2 + bx + c = 0

.

4

Factor the quadratic equation.

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

Factoring allows us to find the potential solutions by setting each factor equal to zero.

5

Find the roots of the equation.

x = 2 \text{ or } x = -1

These are the values of

x

that satisfy the factored quadratic equation.

6

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

Substituting

x = -1

results in division by zero in the terms

\frac{2}{x+1}

and

\frac{8}{x^2-2x-3}

, so

x = -1

is extraneous. Substituting

x = 2

yields a valid statement:

-\frac{8}{3} = -\frac{8}{3}

.

Solutions that make any denominator in the original rational equation equal to zero are extraneous and must be excluded.

Key Concept

Solving rational equations and identifying extraneous solutions.

Question 357Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(2, -5)$ . If the graph passes through the point $(5, 13)$ , what is the value of $f(-1)$ ?

-23

-3

13

-11

Show answer & explanation

Answer: 13

Answer

13

The vertex of the parabola is

(2, -5)

, which indicates that the axis of symmetry is the vertical line

x = 2

. The given point has an

x

-coordinate of

5

, which is a distance of

3

units from the axis of symmetry (

5 - 2 = 3

). The target point has an

x

-coordinate of

-1

, which is also a distance of

3

units from the axis of symmetry (

2 - (-1) = 3

). Because a parabola is perfectly symmetric about its axis of symmetry, any two points that are the same horizontal distance from this line must have the same

y

-coordinate. Thus,

f(-1)

must be equal to

f(5)

, which is

13

. Alternatively, one can find the specific equation of the quadratic function by substituting the vertex and the point

(5, 13)

into the vertex form

f(x) = a(x - 2)^2 - 5

, yielding

a = 2

. Evaluating

f(-1) = 2(-1 - 2)^2 - 5

gives

13

.

Step-by-Step Solution

1

Identify the axis of symmetry from the given vertex.

The axis of symmetry is

x = 2

.

For any quadratic function with a vertex at

(h, k)

, the vertical line

x = h

is the axis of symmetry of its parabolic graph.

2

Determine the horizontal distance from the axis of symmetry to the given point

x = 5

and the target point

x = -1

.

The distance for

x = 5

is

5 - 2 = 3

units. The distance for

x = -1

is

2 - (-1) = 3

units.

Checking if the two

x

-coordinates are symmetric with respect to the line

x = 2

allows us to use the symmetry property of parabolas.

3

Apply the symmetry property to find the function value.

Since both

x = 5

and

x = -1

are equidistant from the axis of symmetry, their function values are equal:

f(-1) = f(5) = 13

.

Points on a parabola that are equidistant from the axis of symmetry have the same

y

-coordinate.

Key Concept

Symmetry of quadratic functions about their vertex axis of symmetry

Question 358Question →

A cubic polynomial function $p$ with integer coefficients has exactly two $x$ -intercepts at $(-2, 0)$ and $(1, 0)$ in the $xy$ -plane. The graph of $y = p(x)$ is tangent to the $x$ -axis at one of these intercepts and intersects the $y$ -axis at $(0, -6)$ . What is the remainder when $p(x)$ is divided by $x + 3$ ?

48

-48

-6

-60

Show answer & explanation

Answer: 48

Answer

48

The correct answer is 48. A cubic polynomial

p(x)

with exactly two

x

-intercepts at

(-2, 0)

and

(1, 0)

that is tangent to the

x

-axis at one of these intercepts must have one factor of multiplicity 2. This gives two possible forms:

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

or

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

. Using the

y

-intercept

(0, -6)

, we can solve for the coefficients: if

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

, then

p(0) = 2a = -6

, which gives

a = -3

. If

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

, then

p(0) = -4b = -6

, which gives

b = 1.5

. Since the polynomial must have integer coefficients, the correct function is

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

. By the Remainder Theorem, dividing

p(x)

by

x + 3

yields a remainder equal to

p(-3)

. Substituting -3 into the function gives

p(-3) = -3(-3+2)(-3-1)^2 = -3(-1)(16) = 48

.

Step-by-Step Solution

1

Identify the general form of the cubic polynomial based on its x-intercepts and their multiplicities.

The polynomial must be of the form

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

or

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

.

Since there are exactly two x-intercepts and the graph is tangent to the x-axis at one of them, one of the factors must have a multiplicity of 2.

2

Determine the coefficients using the y-intercept (0, -6) and check which function has integer coefficients.

Evaluating at x = 0 gives either

2a = -6 \Rightarrow a = -3

or

-4b = -6 \Rightarrow b = 1.5

. Thus, the correct polynomial with integer coefficients is

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

.

The y-intercept gives the value of the function at x = 0. The constraint requires integer coefficients, which rules out b = 1.5.

3

Apply the Remainder Theorem to find the remainder when p(x) is divided by x + 3.

The remainder is

p(-3) = -3(-3+2)(-3-1)^2 = -3(-1)(-4)^2 = 48

.

According to the Remainder Theorem, the remainder of a polynomial p(x) divided by x - c is p(c). Here, c = -3.

Key Concept

Polynomial Factors and Graphs

Estimated Time:2m 0s

Question 359Question →

The mass of a sample of a chemical compound in a reaction decays exponentially. The mass, in grams, of the sample $t$ hours after the reaction starts can be modeled by the function $M(t) = a \cdot b^t$ , where $a$ and $b$ are positive constants. If the mass of the sample is $18$ grams after $2$ hours and $8$ grams after $4$ hours, what is the initial mass, in grams, of the sample?

Show answer & explanation

Answer: 40.5

Answer

40.5

The initial mass of the sample is

40.5

grams (which can also be entered as the fraction

81/2

). This is found by setting up the two equations from the given points:

a \cdot b^2 = 18

and

a \cdot b^4 = 8

. Dividing the second equation by the first eliminates

a

and gives

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

. Substituting

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

back into the first equation yields

a \cdot \frac{4}{9} = 18

. Multiplying both sides by

\frac{9}{4}

results in

a = 40.5

. Since

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

, the initial mass of the sample is

40.5

grams.

Step-by-Step Solution

1

Set up the system of exponential equations using the given coordinates.

a \cdot b^2 = 18

and

a \cdot b^4 = 8

This represents the mass of the sample at

t = 2

and

t = 4

using the model

M(t) = a \cdot b^t

.

2

Divide the second equation by the first equation to eliminate the constant

a

and solve for

b^2

.

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

Dividing the equations yields

\frac{a \cdot b^4}{a \cdot b^2} = \frac{8}{18}

, which simplifies to

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

.

3

Substitute the value of

b^2

back into the first equation to solve for the initial mass

a

.

a = 40.5

Substituting

b^2

gives

a \cdot \frac{4}{9} = 18

. Multiplying both sides by

\frac{9}{4}

yields

a = 18 \cdot \frac{9}{4} = 40.5

.

Key Concept

Solving systems of exponential equations to determine the initial value and decay factor.

Question 360Question →

The quadratic equation $x^2 - 8x - 9 = 0$ can be written in the equivalent form $(x - a)^2 - b = 0$ , where $a$ and $b$ are positive constants. What is the value of $a + b$ ?

11

13

29

21

Show answer & explanation

Answer: 29

Answer

The value of

a + b

is 29.

Completing the square on the quadratic equation

x^2 - 8x - 9 = 0

yields

(x - 4)^2 - 25 = 0

. Comparing this to the form

(x - a)^2 - b = 0

shows that

a = 4

and

b = 25

. The sum of these values is

4 + 25 = 29

.

Step-by-Step Solution

1

Identify the coefficient of the linear term and find half of its value.

The coefficient of the linear term

-8x

is

-8

. Half of

-8

is

-4

.

This is the first step in completing the square.

2

Square the value obtained in the first step and add/subtract it in the equation to form a perfect square trinomial.

(-4)^2 = 16

. The equation becomes

(x^2 - 8x + 16) - 16 - 9 = 0

.

Adding and subtracting

16

maintains the equality while allowing us to group the first three terms as a perfect square.

3

Rewrite the perfect square trinomial and combine the remaining constant terms.

(x - 4)^2 - 25 = 0

.

This simplifies the equation into the desired equivalent form

(x - a)^2 - b = 0

.

4

Compare the equation to the target form

(x - a)^2 - b = 0

to identify the constants

a

and

b

, and calculate

a + b

.

a = 4

and

b = 25

. Therefore,

a + b = 4 + 25 = 29

.

This answers the question by finding the sum of the positive constants

a

and

b

.

Key Concept

Completing the square to rewrite a quadratic equation

Advanced Math

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