Advanced Math Practice Questions — SAT — Page 14

Question 261Question →

The polynomial function $p$ is defined by $p(x) = x^4 - 8x^3 + 20x^2 - 16x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ is tangent to the $x$ -axis at two distinct points. What is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. A quartic polynomial that is tangent to the x-axis at two distinct points must have two roots of multiplicity 2. Thus, it can be factored as p(x) = (x - r_1)^2(x - r_2)^2 = [x^2 - (r_1 + r_2)x + r_1 r_2]^2. Setting S = r_1 + r_2 and P = r_1 r_2, we expand this to p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2. Matching this with the given equation x^4 - 8x^3 + 20x^2 - 16x + c shows that -2S = -8, so S = 4. Using the quadratic term, S^2 + 2P = 20, which yields 16 + 2P = 20, so P = 2. Finally, the constant term is c = P^2 = 2^2 = 4.

Step-by-Step Solution

1

Express the polynomial using its roots and multiplicities.

p(x) = (x - r_1)^2(x - r_2)^2

Since the graph of the quartic polynomial is tangent to the x-axis at two distinct points, it must have two real roots, each with a multiplicity of 2.

2

Group the terms and introduce variables for the sum and product of the roots.

p(x) = [x^2 - Sx + P]^2, where S = r_1 + r_2 and P = r_1 r_2

This simplifies expansion by using the symmetric properties of roots.

3

Expand the squared quadratic expression.

p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2

To compare coefficients with the standard form of the given polynomial.

4

Compare the coefficients of the x^3 term.

-2S = -8, which gives S = 4

To determine the sum of the roots from the given cubic coefficient.

5

Compare the coefficients of the x^2 term using the value of S.

S^2 + 2P = 20, which becomes 16 + 2P = 20, leading to P = 2

To find the product of the roots using the quadratic coefficient.

6

Verify the linear coefficient using S and P.

-2SP = -2(4)(2) = -16

To ensure consistency with the given coefficient of x.

7

Determine the constant term c.

c = P^2 = 2^2 = 4

To find the final value of the constant c.

Key Concept

Representing a polynomial function in factored form based on the multiplicity of its roots and using coefficient matching to determine unknown constants.

Question 262Question →

The system of equations consists of a quadratic equation and a linear equation, where

c

is a constant:

\begin{aligned} y &= 2x^2 - 3x + 1 \\ y &= x + c \end{aligned}

If this system has exactly one real solution, what is the value of

c

?

-3

-1

1

3

Show answer & explanation

Answer: $-1$

Answer

The value of the constant

c

is

-1

.

To find the value of

c

for which the system has exactly one real solution, we equate the two equations to get

2x^2 - 3x + 1 = x + c

. Subtracting

x

and

c

from both sides gives the quadratic equation

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

. A quadratic equation has exactly one real solution when its discriminant,

b^2 - 4ac

, is equal to zero. Substituting

a = 2

,

b = -4

, and the constant term

(1 - c)

into the discriminant formula gives

(-4)^2 - 4(2)(1 - c) = 0

, which simplifies to

16 - 8(1 - c) = 0

. Solving for

c

yields

8 + 8c = 0

, or

c = -1

.

Step-by-Step Solution

1

Equate the two expressions for

y

to set up a single equation in terms of

x

.

2x^2 - 3x + 1 = x + c

Since both equations are equal to

y

, their right-hand sides must be equal at any point of intersection.

2

Rearrange the equation into standard quadratic form,

ax^2 + bx + c_0 = 0

.

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

Subtracting

x

and

c

from both sides groups the terms by their powers of

x

.

3

Write the expression for the discriminant of this quadratic equation.

D = (-4)^2 - 4(2)(1 - c)

For a quadratic equation

ax^2 + bx + c_0 = 0

, the discriminant is given by

b^2 - 4ac_0

where

a = 2

,

b = -4

, and the constant term is

1 - c

.

4

Set the discriminant to zero and solve for

c

to find the condition for exactly one real solution.

16 - 8(1 - c) = 0 \implies 16 - 8 + 8c = 0 \implies 8 + 8c = 0 \implies c = -1

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

Key Concept

Determining the number of solutions of a nonlinear system by setting the discriminant of the combined quadratic equation to zero.

Estimated Time:1m 30s

Question 263Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -2x^2 + 12x - 10$ is translated $4$ units to the left and $k$ units up, where $k$ is a constant, to produce the graph of a new quadratic function $g$ . If the graph of $g$ passes through the origin $(0,0)$ , what is the value of $k$ ?

90

10

-6

3

Show answer & explanation

Answer: -6

Answer

The correct answer is

-6

.

The correct answer is

-6

. By completing the square on the original quadratic function, we rewrite

f(x) = -2(x^2 - 6x) - 10

as

f(x) = -2(x-3)^2 + 8

. A translation of

4

units to the left is represented by replacing

x

with

x+4

, and a translation of

k

units up is represented by adding

k

, giving

g(x) = f(x+4) + k = -2(x+1)^2 + 8 + k

. Since the graph of

g

passes through the origin, we substitute

(0,0)

into the equation:

0 = -2(0+1)^2 + 8 + k

, which simplifies to

0 = 6 + k

, yielding

k = -6

.

Step-by-Step Solution

1

Rewrite the function

f(x)

in vertex form by completing the square.

f(x) = -2(x-3)^2 + 8

Converting the standard form

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

to vertex form

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

helps easily identify the vertex of the parabola. First, factor out

-2

from the variable terms:

f(x) = -2(x^2 - 6x) - 10

. To complete the square inside the parentheses, add and subtract

9

(since

(\frac{-6}{2})^2 = 9

):

f(x) = -2(x^2 - 6x + 9 - 9) - 10

. This simplifies to

f(x) = -2((x-3)^2 - 9) - 10 = -2(x-3)^2 + 18 - 10 = -2(x-3)^2 + 8

.

2

Determine the equation of the translated function

g(x)

.

g(x) = -2(x+1)^2 + 8 + k

Translating a function

4

units to the left is represented by replacing

x

with

x + 4

. Translating a function

k

units up is represented by adding

k

to the entire function. Therefore,

g(x) = f(x+4) + k = -2((x+4)-3)^2 + 8 + k = -2(x+1)^2 + 8 + k

.

3

Substitute the point

(0,0)

into

g(x)

to solve for

k

.

k = -6

Since the graph of

g

passes through the origin

(0,0)

, we have

g(0) = 0

. Substituting

x=0

yields

0 = -2(0+1)^2 + 8 + k \Rightarrow 0 = -2(1) + 8 + k \Rightarrow 0 = 6 + k \Rightarrow k = -6

.

Key Concept

Quadratic transformations and translations in the coordinate plane using vertex form.

Question 264Question →

The function $f$ has the property that $f(3) = 11$ . The function $g$ is defined by $g(x) = f(x) - 4$ . What is the value of $g(3)$ ?

Show answer & explanation

Answer: 7

Answer

7

The value of

g(3)

is found by substituting

x = 3

into the equation

g(x) = f(x) - 4

, which results in

g(3) = f(3) - 4

. Substituting the given value of

f(3) = 11

yields

g(3) = 11 - 4

, which simplifies to

7

.

Step-by-Step Solution

1

Substitute

x = 3

into the function definition of

g(x)

.

g(3) = f(3) - 4

To evaluate the function

g

at a specific input, we replace

x

with

3

in the definition

g(x) = f(x) - 4

.

2

Substitute the given value of

f(3) = 11

into the equation.

g(3) = 11 - 4

The problem states that the value of

f(3)

is equal to

11

.

3

Simplify the expression to find the final value.

g(3) = 7

Subtracting

4

from

11

yields

7

.

Key Concept

Applying vertical translations using function notation.

Estimated Time:40s

Question 265Question →

The table below shows several values of the function $f$ .

$x$	$f(x)$
$-3$	$8$
$-1$	$2$
$1$	$5$
$3$	$-1$

The function $g$ is defined by $g(x) = f(x + 2)$ . What is the value of $g(-1)$ ?

8

5

4

0

Show answer & explanation

Answer: 5

Answer

5

To find the value of

g(-1)

, substitute

x = -1

into the equation

g(x) = f(x + 2)

, which yields

g(-1) = f(-1 + 2) = f(1)

. Looking at the table, when the input is

1

, the output of the function

f

is

5

. Therefore, the value of

g(-1)

is

5

.

Step-by-Step Solution

1

Substitute the given input value of

-1

for

x

in the definition of

g(x)

.

g(-1) = f(-1 + 2)

To find

g(-1)

, we need to replace

x

with

-1

in the equation

g(x) = f(x + 2)

.

2

Simplify the expression inside the function notation.

g(-1) = f(1)

Performing the addition

-1 + 2

yields

1

.

3

Use the table to find the value of

f(1)

.

f(1) = 5

Looking at the row in the table where

x = 1

, the corresponding value of

f(x)

is

5

.

Key Concept

Function transformations and notation using tables

Question 266Question →

A cubic polynomial function $p$ is defined by $p(x) = a(x + 3)(x - 1)(x - k)$ , where $a$ and $k$ are constants. In the $xy$ -plane, the $y$ -intercept of the graph of $y = p(x)$ is $(0, 6)$ . The graph of the shifted function $y = p(x - 2)$ passes through the point $(4, -30)$ . What is the third $x$ -intercept of the graph of $p$ ?

-1

9

\frac{1}{3}

-\frac{1}{2}

Show answer & explanation

Answer: $-1$

Answer

-1

The correct answer is

-1

because the

y

-intercept

(0, 6)

gives

3ak = 6

, which simplifies to

ak = 2

. The shift

y = p(x-2)

passing through

(4, -30)

implies

p(2) = -30

. Substituting

x=2

into the polynomial expression yields

5a(2-k) = -30

, which expands to

10a - 5ak = -30

. Substituting

ak = 2

gives

10a - 10 = -30

, solving to

a = -2

. Using

ak = 2

, we find

k = -1

. The third factor is

(x - k) = (x + 1)

, which corresponds to the third

x

-intercept at

x = -1

.

Step-by-Step Solution

1

Use the

y

-intercept of the graph of

y = p(x)

to establish a relationship between

a

and

k

.

p(0) = a(0 + 3)(0 - 1)(0 - k) = 3ak = 6 \Rightarrow ak = 2

.

The

y

-intercept is the point on the graph where

x = 0

.

2

Translate the point on the shifted graph back to the original function

p(x)

.

p(4 - 2) = p(2) = -30

.

Since the graph of

y = p(x - 2)

passes through

(4, -30)

, substituting

x = 4

yields

y = -30

.

3

Substitute

x = 2

into the expression for

p(x)

to set up the second equation.

p(2) = a(2 + 3)(2 - 1)(2 - k) = 5a(2 - k) = 10a - 5ak = -30

.

This establishes a system of equations with the relation from Step 1.

4

Solve the system of equations by substituting

ak = 2

into the second equation.

10a - 5(2) = -30 \Rightarrow 10a - 10 = -30 \Rightarrow 10a = -20 \Rightarrow a = -2

. Since

ak = 2

, we have

-2k = 2 \Rightarrow k = -1

.

Solving for the unknown constants

a

and

k

determines the specific polynomial expression.

5

Identify the third

x

-intercept from the factored form of the polynomial.

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

. The factors correspond to roots at

x = -3

,

x = 1

, and

x = -1

. The third

x

-intercept is

-1

.

The third factor

(x - k)

becomes

(x + 1)

when

k = -1

, yielding the root and intercept at

x = -1

.

Key Concept

Polynomial Factors and Graphs

Question 267Question →

An equation is given as follows:

\frac{x}{x - 3} - \frac{4}{x + 2} = \frac{20}{x^2 - x - 6}

What is the only value of $x$ for which this equation is true?

Show answer & explanation

Answer: 4

Answer

4

To solve the equation, factor the denominator on the right side:

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

. The least common denominator is

(x - 3)(x + 2)

. Multiplying both sides by

(x - 3)(x + 2)

clears the fractions, resulting in

x(x + 2) - 4(x - 3) = 20

. Expanding the terms gives

x^2 + 2x - 4x + 12 = 20

. Simplifying and writing this in standard form yields

x^2 - 2x - 8 = 0

. Factoring the quadratic gives

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

, which gives the candidate solutions

x = 4

and

x = -2

. Substituting

x = -2

back into the original equation results in a denominator of zero, so

x = -2

is an extraneous solution. The only valid solution is

x = 4

.

Step-by-Step Solution

1

Factor the quadratic trinomial in the denominator of the right side of the equation.

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

Identifying the factors of the quadratic trinomial helps find the least common denominator of the rational equation.

2

Multiply the entire equation by the least common denominator

(x - 3)(x + 2)

to clear the fractions.

x(x + 2) - 4(x - 3) = 20

, with the constraints that

x \neq 3

and

x \neq -2

.

This simplifies the rational equation into a standard polynomial equation.

3

Distribute and combine like terms to write the equation in standard quadratic form.

x^2 - 2x - 8 = 0

Rewriting the equation in the form

ax^2 + bx + c = 0

is necessary to solve it by factoring.

4

Factor the quadratic equation.

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

Factoring allows us to apply the zero product property to find candidate solutions.

5

Check the candidate solutions

x = 4

and

x = -2

against the original equation to identify extraneous solutions.

x = 4

is the only valid solution because

x = -2

makes the denominators in the original equation equal to zero.

Any solution that makes a denominator in the original rational expression equal to zero is extraneous and must be excluded.

Key Concept

Solving rational equations by finding a common denominator, clearing fractions, and checking for extraneous solutions.

Question 268Question →

If $81^{y-2} = 27^{y+1}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 11

Answer

11

The correct answer is 11. By expressing both sides of the equation with the common base of 3, the equation simplifies from

81^{y-2} = 27^{y+1}

to

(3^4)^{y-2} = (3^3)^{y+1}

. Applying the power of a power rule gives

3^{4y-8} = 3^{3y+3}

. Equating the exponents yields

4y - 8 = 3y + 3

, which solves to

y = 11

.

Step-by-Step Solution

1

Express both bases as powers of 3

(3^4)^{y-2} = (3^3)^{y+1}

To solve an exponential equation algebraically, it is helpful to express both sides using a common base.

2

Apply the power of a power exponent rule

3^{4(y-2)} = 3^{3(y+1)}

The rule

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

allows us to simplify the exponent expressions by multiplying the exponents.

3

Equate the exponents

4(y-2) = 3(y+1)

Since the bases are equal, the exponents must be equal for the equation to hold true.

4

Distribute the coefficients

4y - 8 = 3y + 3

Expanding the linear expressions prepares the equation for isolation of the variable.

5

Solve the linear equation for

y

y = 11

Subtract

3y

from both sides to get

y - 8 = 3

, then add 8 to both sides to isolate

y

.

Key Concept

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

Alternative Method

Alternatively, substitute the value of 11 back into the original equation to verify that both sides are equal:

81^{11-2} = 81^9 = (3^4)^9 = 3^{36}

and

27^{11+1} = 27^{12} = (3^3)^{12} = 3^{36}

.

Estimated Time:1m 30s

Question 269Question →

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

\frac{x^2 - 3x}{x^2 - 3x - 2} + \frac{x^2 - 3x - 2}{x^2 - 3x} = \frac{5}{2}

?

Show answer & explanation

Answer: 7

Answer

The sum of all positive real solutions is 7.

By substituting

u = x^2 - 3x

, the original rational equation simplifies to

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

. Multiplying both sides by the common denominator

2u(u-2)

and simplifying results in the quadratic equation

u^2 - 2u - 8 = 0

. Factoring gives

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

, so

u = 4

or

u = -2

. Substituting back

x^2 - 3x

for

u

leads to two quadratic equations:

x^2 - 3x = 4

(which has solutions

x = 4

and

x = -1

) and

x^2 - 3x = -2

(which has solutions

x = 2

and

x = 1

). Checking the denominators, none of these solutions make the original denominators zero, so they are all valid. The positive solutions are

1

,

2

, and

4

, and their sum is

1 + 2 + 4 = 7

.

Step-by-Step Solution

1

Introduce a substitution variable to simplify the rational equation.

Letting

u = x^2 - 3x

transforms the equation into

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

.

This reduces the degree of the rational expression and simplifies the algebraic manipulation required to solve it.

2

Eliminate the denominators by multiplying by the least common denominator.

Multiplying by

2u(u-2)

gives

2u^2 + 2(u-2)^2 = 5u(u-2)

, which simplifies to

u^2 - 2u - 8 = 0

.

This converts the rational equation into a standard quadratic equation in terms of

u

.

3

Solve the quadratic equation for

u

by factoring.

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

, which gives

u = 4

or

u = -2

.

Finding the values of

u

allows us to set up equations to solve for the original variable

x

.

4

Substitute back

x^2 - 3x

for

u

and solve the resulting quadratic equations for

x

.

From

x^2 - 3x = 4

, we get

(x-4)(x+1) = 0 \implies x = 4, -1

. From

x^2 - 3x = -2

, we get

(x-2)(x-1) = 0 \implies x = 2, 1

.

This yields all real values of

x

that satisfy the original algebraic structure.

5

Filter for positive real solutions and calculate their sum.

The positive solutions are

1

,

2

, and

4

. Their sum is

1 + 2 + 4 = 7

.

The question specifically asks for the sum of only the positive real solutions.

Key Concept

Solving rational equations using algebraic substitution and factoring quadratic equations.

Alternative Method

Instead of using substitution directly, the equation can be solved by multiplying by the common denominator

(x^2 - 3x - 2)(x^2 - 3x)

to get a fourth-degree polynomial:

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

. Letting

z = x^2 - 3x

at this stage simplifies this expression to

2z^2 + 2(z-2)^2 = 5z(z-2)

, which avoids full expansion into a fourth-degree polynomial and leads to the same quadratic in

z

.

Estimated Time:3m 0s

Question 270Question →

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

3

7

8

9

Show answer & explanation

Answer: 7

Answer

The correct value of

x

is 7.

To solve the equation

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

, write all bases as powers of 3:

9 = 3^2

and

27 = 3^3

. Substituting these into the equation gives

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

. Applying the power rule of exponents, we get

3^{4x - 2} = \frac{3^{3x + 6}}{3^1}

. Using the quotient rule of exponents on the right side, we subtract the exponent in the denominator (which is 1) from the exponent in the numerator:

3^{3x + 6 - 1} = 3^{3x + 5}

. Now we have

3^{4x - 2} = 3^{3x + 5}

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

4x - 2 = 3x + 5

. Solving for

x

by subtracting

3x

and adding 2 to both sides gives the correct value of 7.

Step-by-Step Solution

1

Express all bases as powers of 3.

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

To solve exponential equations with different bases, we rewrite them using a common base to apply exponent laws.

2

Apply the power of a power rule to simplify exponents.

3^{4x - 2} = \frac{3^{3x + 6}}{3^1}

Multiply exponents when raising a power to another power:

(a^m)^n = a^{m \cdot n}

.

3

Apply the quotient rule to simplify the right side of the equation.

3^{4x - 2} = 3^{3x + 5}

Subtract the exponent of the denominator from the numerator when dividing:

\frac{a^m}{a^n} = a^{m-n}

.

4

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

x = 7

Since the bases on both sides are equal, their exponents must be equal:

4x - 2 = 3x + 5

.

Key Concept

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

Question 271Question →

The function $f$ is defined by $f(x) = (x - 4)(x - 2)(x + k)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane is $(0, 24)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

3

The

y

-intercept of the graph of

y = f(x)

is the point where

x = 0

. Since the

y

-intercept is

(0, 24)

, it follows that

f(0) = 24

. Substituting

0

for

x

in the equation for

f(x)

gives

f(0) = (0 - 4)(0 - 2)(0 + k) = (-4)(-2)(k) = 8k

. Setting this equal to the

y

-value of the intercept yields

8k = 24

. Dividing both sides of the equation by

8

gives

k = 3

.

Step-by-Step Solution

1

Use the definition of the

y

-intercept to find the value of

f(0)

f(0) = 24

The

y

-intercept of a graph is the point where the graph crosses the

y

-axis, corresponding to

x = 0

. Given the point

(0, 24)

,

f(0)

must equal

24

.

2

Evaluate the polynomial at

x = 0

in terms of

k

f(0) = 8k

Substituting

0

for

x

in

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

gives

f(0) = (-4)(-2)(k)

, which simplifies to

8k

.

3

Set the evaluated expression equal to the

y

-intercept value and solve for

k

k = 3

Equating

8k

to

24

and dividing both sides by

8

yields

k = 3

.

Key Concept

Evaluating a factored polynomial function at

x = 0

determines its

y

-intercept. Using a given

y

-intercept allows solving for unknown coefficients or constants within the factors of the polynomial.

Question 272Question →

In the quadratic equation $x^2 + kx + (k + 3) = 0$ , $k$ is a constant. If the equation has exactly one real solution, which of the following is a possible value of $k$ ?

-6

3

6

2

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. For the quadratic equation

x^2 + kx + (k + 3) = 0

to have exactly one real solution, its discriminant must be equal to zero. Setting the discriminant

b^2 - 4ac

to zero gives

k^2 - 4(1)(k + 3) = 0

, which simplifies to

k^2 - 4k - 12 = 0

. Factoring this equation yields

(k - 6)(k + 2) = 0

, meaning the possible values of

k

are

6

and

-2

. Among the choices, 6 is the only possible value listed.

Step-by-Step Solution

1

Identify the coefficients of the given quadratic equation

x^2 + kx + (k + 3) = 0

.

The coefficients are

a = 1

,

b = k

, and

c = k + 3

.

To apply the discriminant formula, we need to identify the standard form coefficients

a

,

b

, and

c

.

2

Set the discriminant of the quadratic equation to zero.

The discriminant is

D = b^2 - 4ac

. Setting

D = 0

gives

k^2 - 4(1)(k + 3) = 0

, which simplifies to

k^2 - 4k - 12 = 0

.

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

3

Solve the quadratic equation

k^2 - 4k - 12 = 0

for

k

.

Factoring the quadratic yields

(k - 6)(k + 2) = 0

. Therefore, the possible values of

k

are

6

and

-2

.

Solving the equation reveals the values of the constant

k

that satisfy the condition.

Key Concept

Using the discriminant of a quadratic equation to determine the number of real solutions.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

k^2 - 4k - 12 = 0

, where

k = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)} = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm 8}{2}

, giving

k = 6

or

k = -2

.

Estimated Time:1m 30s

Question 273Question →

In the equation below,

x

is a real number.

\sqrt{7 - 6x} = x + 8

What is the value of

x + 5

?

8

-14

2

-3

Show answer & explanation

Answer: 2

Answer

The correct answer is the value 2.

Substituting the only valid solution of the radical equation, which is

-3

, into the expression

x + 5

yields

2

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical sign.

7 - 6x = (x + 8)^2

, which expands to

7 - 6x = x^2 + 16x + 64

.

Squaring both sides of a radical equation allows it to be rewritten as a standard polynomial equation.

2

Rearrange the terms to set the quadratic equation equal to zero.

x^2 + 22x + 57 = 0

Grouping all terms on one side of the equation puts it in standard quadratic form.

3

Factor the quadratic equation to find potential solutions for

x

.

(x + 19)(x + 3) = 0

, which gives the potential solutions

x = -19

and

x = -3

.

Factoring allows us to solve the quadratic equation easily.

4

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

Substituting

x = -3

yields

\sqrt{7 - 6(-3)} = -3 + 8 \Rightarrow \sqrt{25} = 5

, which is true. Substituting

x = -19

yields

\sqrt{7 - 6(-19)} = -19 + 8 \Rightarrow \sqrt{121} = -11

, which is false because a principal square root must be non-negative. Therefore,

x = -3

is the only valid solution.

Squaring both sides can introduce extraneous solutions that must be discarded.

5

Substitute the valid solution

x = -3

into the target expression

x + 5

.

-3 + 5 = 2

The question asks for the value of the expression

x + 5

.

Key Concept

Solving radical equations by squaring and checking for extraneous solutions.

Question 274Question →

For the function $f$ , it is given that $f(4) = 18$ . The function $g$ is defined by $g(x) = \frac{1}{3}f(x + 2)$ . What is the value of $g(2)$ ?

Show answer & explanation

Answer: 6

Answer

The value of

g(2)

is 6.

To evaluate

g(2)

, substitute

x = 2

into the definition

g(x) = \frac{1}{3}f(x+2)

, which yields

g(2) = \frac{1}{3}f(2+2) = \frac{1}{3}f(4)

. Since

f(4) = 18

, this simplifies to

\frac{1}{3}(18) = 6

.

Step-by-Step Solution

1

Substitute

x = 2

into the expression for

g(x)

.

g(2) = \frac{1}{3}f(2 + 2)

To evaluate the function

g

at

x = 2

, we substitute 2 for every occurrence of

x

in the function definition.

2

Simplify the input argument for the function

f

.

g(2) = \frac{1}{3}f(4)

Adding 2 and 2 inside the parentheses simplifies the input of

f

to 4.

3

Substitute the given value of

f(4)

into the simplified expression.

g(2) = \frac{1}{3}(18)

The problem states that

f(4) = 18

.

4

Perform the final multiplication.

6

Multiplying 18 by

\frac{1}{3}

is equivalent to dividing 18 by 3, which yields 6.

Key Concept

Evaluating a transformed function by substituting a value into function notation.

Estimated Time:45s

Question 275Question →

Consider the system of equations below:

\begin{aligned} y &= 3x^2 - 5x - 4 \\ y &= x^2 - 2x + 5 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $x > 0$ , what is the value of $y$ ?

Show answer & explanation

Answer: 8

Answer

The value of

y

is 8.

By setting the two equations equal to each other, we obtain

3x^2 - 5x - 4 = x^2 - 2x + 5

. Simplifying this equation by moving all terms to one side yields

2x^2 - 3x - 9 = 0

. Factoring this quadratic equation gives

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

, which has solutions

x = -1.5

and

x = 3

. Since the problem specifies that

x > 0

, we must use

x = 3

. Substituting

x = 3

into the second equation, we find

y = (3)^2 - 2(3) + 5 = 8

. Substituting into the first equation also yields

y = 3(3)^2 - 5(3) - 4 = 8

. Therefore, the value of

y

is 8.

Step-by-Step Solution

1

Set the quadratic expressions equal to each other.

3x^2 - 5x - 4 = x^2 - 2x + 5

Since both equations are solved for

y

, their right-hand sides must be equal at any point of intersection.

2

Rearrange the terms to set the quadratic equation to zero.

2x^2 - 3x - 9 = 0

Putting the equation in standard form

ax^2 + bx + c = 0

allows us to solve it by factoring.

3

Factor the quadratic expression to find the roots.

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

, which yields

x = -1.5

or

x = 3

.

Factoring shows the values of

x

that satisfy the system.

4

Select the positive root and substitute it back to find

y

.

y = 8

The problem specifies

x > 0

, so we use

x = 3

. Substituting

x = 3

into

y = x^2 - 2x + 5

gives the corresponding

y

-value.

Key Concept

Solving a system of nonlinear equations by setting the equations equal to each other and solving the resulting quadratic equation.

Question 276Question →

An environmental study monitors the populations of two different plant species in a conservation area. The population of Species A is modeled by a linear function, $A(t) = 120 + 15t$ , where $t$ represents the number of years since the start of the study. The population of Species B is modeled by an exponential function, $B(t) = c \cdot d^t$ , where $c$ and $d$ are constants. At the start of the study ( $t = 0$ ), the population of Species A is $4$ times the population of Species B. After $2$ years, the population of Species A is equal to the population of Species B. What is the population of Species B after $4$ years?

Show answer & explanation

Answer: 750

Answer

The population of Species B after

4

years is

750

.

To find the population of Species B after

4

years, we evaluate the models at the given points. At

t = 0

,

A(0) = 120

. Since the population of Species A is

4

times that of Species B at

t = 0

, the initial population of Species B is

30

, which gives

c = 30

. At

t = 2

,

A(2) = 120 + 15(2) = 150

. Since the populations are equal at

t = 2

, we have

B(2) = 30 \cdot d^2 = 150

, which simplifies to

d^2 = 5

. The population of Species B at

t = 4

is given by

B(4) = 30 \cdot d^4 = 30 \cdot (d^2)^2 = 30 \cdot 5^2 = 750

.

Step-by-Step Solution

1

Find the population of Species A at the start of the study (

t = 0

)

A(0) = 120

This establishes the baseline population of Species A to find the corresponding initial population of Species B.

2

Determine the constant

c

, which represents the initial population of Species B

c = 30

Since the population of Species A is

4

times that of Species B at

t = 0

, we solve

120 = 4c

.

3

Calculate the population of Species A after

2

years (

t = 2

)

A(2) = 150

This value is needed because the population of Species B equals the population of Species A at

t = 2

.

4

Solve for the growth factor term

d^2

d^2 = 5

Using the equality

B(2) = 150

, we solve

30 \cdot d^2 = 150

.

5

Calculate the population of Species B after

4

years (

t = 4

)

B(4) = 750

Using the exponential model

B(t) = 30 \cdot d^t

, we find

B(4) = 30 \cdot d^4 = 30 \cdot (d^2)^2 = 30 \cdot 5^2

.

Key Concept

Solving systems involving linear and exponential models using initial conditions and key points.

Question 277Question →

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 $(4, 9)$ . What is the positive difference between the two $x$ -intercepts of the graph?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

The quadratic function can be represented in vertex form as

f(x) = -(x - 4)^2 + 9

since the leading coefficient is

-1

and the vertex is at

(4, 9)

. Setting the function equal to zero to find the

x

-intercepts gives

(x - 4)^2 = 9

, which yields

x = 7

and

x = 1

. The positive difference between these intercepts is

7 - 1 = 6

.

Step-by-Step Solution

1

Write the quadratic function in vertex form using the given vertex

(4, 9)

and the leading coefficient.

f(x) = -(x - 4)^2 + 9

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Since the coefficient of

x^2

is

-1

, we have

a = -1

,

h = 4

, and

k = 9

.

2

Set

f(x) = 0

to find the

x

-intercepts.

(x - 4)^2 = 9

The

x

-intercepts of a graph are the points where

f(x) = 0

.

3

Solve for

x

.

x = 7

and

x = 1

Taking the square root of both sides gives

x - 4 = \pm 3

, which results in

x = 7

and

x = 1

.

4

Find the positive difference between the two

x

-intercepts.

6

Subtract the smaller

x

-intercept from the larger

x

-intercept:

7 - 1 = 6

.

Key Concept

Finding the intercepts of a quadratic function using its vertex form

Alternative Method

Alternatively, expand the vertex form

f(x) = -(x - 4)^2 + 9

to get

f(x) = -(x^2 - 8x + 16) + 9 = -x^2 + 8x - 7

. Factoring this expression gives

f(x) = -(x - 7)(x - 1)

. The roots are

x = 7

and

x = 1

, and their difference is

7 - 1 = 6

.

Estimated Time:1m 30s

Question 278Question →

A parabola in the $xy$ -plane has vertex $(2, 11)$ and passes through the point $(5, -7)$ . The equation of the parabola is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Show answer & explanation

Answer: 9

Answer

The value of

a + b + c

is

9

.

The correct value of

a + b + c

is

9

. This is found by writing the parabola's equation in vertex form as

y = -2(x - 2)^2 + 11

and expanding it to standard form

y = -2x^2 + 8x + 3

, which gives the coefficients

a = -2

,

b = 8

, and

c = 3

. Alternatively, substituting

x = 1

directly into the vertex form gives

f(1) = a(1)^2 + b(1) + c = -2(1 - 2)^2 + 11 = 9

.

Step-by-Step Solution

1

Write the equation of the parabola in vertex form using the given vertex

(2, 11)

.

y = a(x - 2)^2 + 11

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex of the parabola.

2

Substitute the coordinates of the point

(5, -7)

into the vertex form equation to solve for the constant

a

.

a = -2

Since the point

(5, -7)

lies on the parabola, substituting

x = 5

and

y = -7

allows us to solve the linear equation

-7 = a(5 - 2)^2 + 11

for

a

.

3

Expand the vertex form equation

y = -2(x - 2)^2 + 11

into standard form

y = ax^2 + bx + c

to identify the coefficients

a

,

b

, and

c

.

y = -2x^2 + 8x + 3

, which gives

a = -2

,

b = 8

, and

c = 3

.

Expanding the squared term and distributing the coefficient

-2

converts the equation to standard form, making it easy to read off the coefficients.

4

Calculate the sum of the coefficients

a + b + c

.

a + b + c = 9

Adding the identified coefficients:

-2 + 8 + 3 = 9

.

Key Concept

Writing and converting quadratic functions between vertex form

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

and standard form

y = ax^2 + bx + c

.

Alternative Method

Instead of expanding the vertex form equation to find the individual coefficients

a

,

b

, and

c

, recognize that the expression

a + b + c

is equal to

f(1)

for the quadratic function

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

. After finding

a = -2

using the vertex form

f(x) = a(x - 2)^2 + 11

, directly evaluate

f(1) = -2(1 - 2)^2 + 11 = 9

.

Estimated Time:2m 30s

Question 279Question →

The polynomial function $p$ is defined by $p(x) = x^3 + 4x^2 - 7x - 10$ . If $p(2) = 0$ , which of the following expressions must be a factor of $p(x)$ ?

x + 2

x - 2

x - 1

x - 5

Show answer & explanation

Answer: $x - 2$

Answer

The expression

x - 2

must be a factor of

p(x)

.

According to the Factor Theorem, if a polynomial

p(x)

evaluates to

0

at a certain value

x = c

(meaning

p(c) = 0

), then

(x - c)

is a factor of the polynomial. Since we are given that

p(2) = 0

, the value

2

is a root of

p(x)

, which means the expression

x - 2

must be a factor of the polynomial.

Step-by-Step Solution

1

Identify the given root of the polynomial.

The problem states that

p(2) = 0

, which means that

x = 2

is a root of the polynomial

p(x)

.

A root is any value of

x

for which the polynomial evaluates to

0

.

2

Apply the Factor Theorem to write the corresponding factor.

According to the Factor Theorem, if

c

is a root of a polynomial

p(x)

, then

(x - c)

is a factor of

p(x)

. Substituting

c = 2

gives

(x - 2)

as a factor.

The Factor Theorem directly relates the roots of a polynomial to its linear factors.

Key Concept

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

.

Question 280Question →

The graph of the function $f$ in the $xy$ -plane passes through the point $(5, -2)$ . The function $g$ is defined by $g(x) = f(x) + 6$ . Which of the following points must lie on the graph of $g$ ?

(5, -8)

(11, -2)

(5, 4)

(-1, -2)

Show answer & explanation

Answer: $(5, 4)$

Answer

The point

(5, 4)

Since the point

(5, -2)

is on the graph of

f

, we know that

f(5) = -2

. The function

g

is defined by

g(x) = f(x) + 6

, which represents a vertical shift upward by

6

units. Substituting

x = 5

into the equation for

g

yields

g(5) = f(5) + 6 = -2 + 6 = 4

. Therefore, the point

(5, 4)

must be on the graph of

g

.

Step-by-Step Solution

1

Determine the value of

f(5)

using the given point on the graph of

f

.

f(5) = -2

Since the point

(5, -2)

lies on the graph of

f

, the input

x = 5

corresponds to the output

y = -2

.

2

Use the definition of

g(x)

to evaluate

g(5)

.

g(5) = f(5) + 6 = -2 + 6 = 4

Substitute

x = 5

into the definition

g(x) = f(x) + 6

to find the corresponding

y

-value for the graph of

g

.

Key Concept

Vertical translations of function graphs

Estimated Time:45s

Advanced Math

Answer

Step-by-Step Solution

Key Concept

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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