All practice questions

If $x$ is the real solution to the equation below, what is the value of $x$ ?

\frac{2x^2 - 30x + 100}{x - 5} + \frac{140}{x + 2} = 10

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

To solve the rational equation, we first state the domain restrictions as

x \neq 5

and

x \neq -2

because these values make the denominators zero. Next, we factor the numerator of the first term:

2x^2 - 30x + 100 = 2(x^2 - 15x + 50) = 2(x - 5)(x - 10)

. Since

x \neq 5

, we can cancel the common factor of

x - 5

from the numerator and denominator, which simplifies the first term to

2(x - 10) = 2x - 20

. The equation can then be rewritten as

2x - 20 + \frac{140}{x + 2} = 10

. Subtracting 10 from both sides yields

2x - 30 + \frac{140}{x + 2} = 0

. Dividing the entire equation by 2 simplifies the coefficients, giving

x - 15 + \frac{70}{x + 2} = 0

. To clear the fraction, we multiply the entire equation by

x + 2

, leading to

(x - 15)(x + 2) + 70 = 0

. Expanding the product gives

x^2 - 13x - 30 + 70 = 0

, which simplifies to

x^2 - 13x + 40 = 0

. Factoring this quadratic equation yields

(x - 5)(x - 8) = 0

, giving the potential solutions

x = 5

and

x = 8

. However,

x = 5

is extraneous because it is restricted from the domain of the original rational expression. Therefore, the only real solution is 8.

Step-by-Step Solution

1

Find the values of

x

that make any denominator equal to zero to identify domain restrictions.

x \neq 5

and

x \neq -2

Dividing by zero is undefined in real numbers, so these values are excluded from the domain.

2

Factor the numerator of the first term,

2x^2 - 30x + 100

.

2(x - 5)(x - 10)

Factoring allows us to check for common factors in the numerator and denominator.

3

Simplify the first rational term by dividing the numerator and denominator by the common factor

x - 5

, under the assumption that

x \neq 5

.

2(x - 10) = 2x - 20

Reducing the expression simplifies the algebraic structure of the equation.

4

Substitute the simplified expression back into the original equation and subtract 10 from both sides.

2x - 30 + \frac{140}{x + 2} = 0

To consolidate the constant terms and prepare the equation for clearing the remaining fraction.

5

Divide the entire equation by 2.

x - 15 + \frac{70}{x + 2} = 0

Dividing by the common factor simplifies the coefficients before expanding.

6

Multiply the entire equation by

x + 2

to clear the denominator, then expand and combine like terms.

x^2 - 13x + 40 = 0

This transforms the rational equation into a standard quadratic equation.

7

Factor the quadratic equation.

(x - 5)(x - 8) = 0

Factoring is the most direct method to find the roots of the quadratic equation.

8

Verify the candidate solutions against the domain restrictions identified in Step 1.

The candidate solution

x = 5

is extraneous because it is excluded from the domain of the original equation. The candidate solution

x = 8

is valid.

Extraneous solutions introduced during algebraic steps must be excluded from the final solution set.

Key Concept

Solving rational equations by simplifying terms, finding common denominators, and identifying extraneous solutions.

Question 1762Question →

In the $xy$ -plane, the system of equations below has exactly one real solution.

y = -x^2 + 6x + 2

y + 2x = k

If $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

18

To find the value of the constant

k

for which the system of equations has exactly one real solution, we can solve the system by substitution. Substituting

y = k - 2x

from the second equation into the first equation gives

k - 2x = -x^2 + 6x + 2

. Rearranging the terms to write this quadratic equation in standard form,

ax^2 + bx + c = 0

, yields

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

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

b^2 - 4ac

, is equal to zero. Substituting

a = 1

,

b = -8

, and

c = k - 2

into the discriminant formula gives

(-8)^2 - 4(1)(k - 2) = 0

. Simplifying this expression results in

64 - 4k + 8 = 0

, which simplifies further to

72 - 4k = 0

. Solving for

k

yields

k = 18

.

Step-by-Step Solution

1

Rewrite the linear equation to express

y

in terms of

x

and

k

.

y = k - 2x

This allows substituting the expression for

y

into the quadratic equation to eliminate

y

.

2

Substitute the expression for

y

into the quadratic equation.

k - 2x = -x^2 + 6x + 2

To construct a single quadratic equation in terms of

x

whose solutions correspond to the system's intersection points.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

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

To identify the coefficients

a = 1

,

b = -8

, and

c = k - 2

.

4

Set the discriminant of the quadratic equation to zero.

(-8)^2 - 4(1)(k - 2) = 0

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

5

Solve the linear equation for

k

.

k = 18

Simplify the equation:

64 - 4k + 8 = 0 \implies 72 - 4k = 0 \implies 4k = 72 \implies k = 18

.

Key Concept

Nonlinear Systems of Equations

Question 1763Question →

The table below shows the population of a colony of bacteria, $P(t)$ , at various times $t$ , in hours.

Time $t$ (hours)	Population $P(t)$
0	120
3	960
6	7,680

If the population can be modeled by an exponential function of the form $P(t) = a \cdot b^t$ , where $a$ and $b$ are constants, what is the value of $b$ ?

2

8

4

\frac{8}{3}

Show answer & explanation

Answer: 2

Answer

2

The correct answer is the value that represents the hourly growth factor of the bacteria. Since the initial population at

t = 0

is 120, we have

a = 120

. Using the population at

t = 3

, we set up the equation

120 \cdot b^3 = 960

. Dividing both sides by 120 yields

b^3 = 8

. Taking the cube root of 8 gives

b = 2

. We can verify this with the third data point:

120 \cdot 2^6 = 120 \cdot 64 = 7,680

, which matches the table.

Step-by-Step Solution

1

Find the value of the constant

a

using the initial value from the table.

a = 120

When

t = 0

, the population

P(0) = 120

. Substituting these values into

P(t) = a \cdot b^t

gives

120 = a \cdot b^0

, and since

b^0 = 1

, we get

a = 120

.

2

Set up an equation to solve for

b

using another data point from the table.

120 \cdot b^3 = 960

When

t = 3

, the population

P(3) = 960

. Substituting

a = 120

,

t = 3

, and

P(3) = 960

into the function gives

960 = 120 \cdot b^3

.

3

Isolate

b^3

and solve for

b

by taking the cube root.

b = 2

Dividing both sides of

120 \cdot b^3 = 960

by

120

gives

b^3 = 8

. Taking the cube root of both sides gives

b = \sqrt[3]{8} = 2

.

Key Concept

Determining parameters of exponential functions from a table of values.

Question 1764Question →

A store owner purchases an item from a wholesaler. She increases the wholesaler's cost by $40\%$ to set the retail price of the item. During a clearance sale, the retail price is discounted by $20\%$ . If the clearance sale price of the item is $84.00$ dollars, what was the wholesaler's cost of the item, in dollars?

70

75

63

72

Show answer & explanation

Answer: 75

Answer

75

The correct answer is

75

. The relationship between the wholesaler's cost

C

, the retail price, and the clearance price is

C \times 1.40 \times 0.80 = 84

. Simplifying the expression yields

1.12C = 84

, which solves to

C = 75

.

Step-by-Step Solution

1

Represent the retail price in terms of the wholesaler's cost

C

.

Retail price =

1.40C

The retail price represents a

40\%

markup on the original wholesaler's cost.

2

Represent the clearance sale price in terms of the retail price.

Clearance price =

0.80 \times (1.40C) = 1.12C

A

20\%

discount reduces the price to

80\%

of the retail price.

3

Set up the equation using the given clearance sale price of

84.00

dollars and solve for

C

.

1.12C = 84 \implies C = \frac{84}{1.12} = 75

Dividing the sale price by the net percentage factor isolates the initial cost.

Key Concept

Applying consecutive percentage changes to find an unknown initial value.

Estimated Time:1m 30s

Question 1765Question →

If $\frac{\left(\frac{1}{4}\right)^{2x-5}}{8^{3-x}} = \left(\sqrt[3]{2}\right)^{x-5}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

2

By writing all terms with a common base of 2, the equation simplifies to

2^{-x+1} = 2^{\frac{x-5}{3}}

. Equating the exponents yields

-x + 1 = \frac{x-5}{3}

. Solving this linear equation gives

x = 2

.

Step-by-Step Solution

1

Rewrite each base in terms of base 2.

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

,

8 = 2^3

, and

\sqrt[3]{2} = 2^{\frac{1}{3}}

Expressing all terms with a common base allows the exponents to be directly compared.

2

Substitute the bases back into the left side of the equation and simplify using exponent rules.

\frac{(2^{-2})^{2x-5}}{(2^3)^{3-x}} = \frac{2^{-4x+10}}{2^{9-3x}} = 2^{-x+1}

Applying the power rule

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

and quotient rule

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

simplifies the fraction.

3

Rewrite the right side of the equation using base 2.

(\sqrt[3]{2})^{x-5} = (2^{\frac{1}{3}})^{x-5} = 2^{\frac{x-5}{3}}

Applying the power rule to the fractional exponent converts the radical expression.

4

Equate the simplified left and right sides, then set the exponents equal to each other.

-x + 1 = \frac{x-5}{3}

If two exponential expressions with the same base are equal, their exponents must be equal.

5

Solve the linear equation for

x

.

-3x + 3 = x - 5 \implies 8 = 4x \implies x = 2

Isolating the variable

x

yields the final solution.

Key Concept

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

Question 1766Question →

If one of the solutions to the equation $2x^2 - 9x + c = 0$ , where $c$ is a constant, is $x = 2$ , what is the other solution to the equation?

-\frac{13}{2}

-\frac{5}{2}

\frac{5}{2}

7

Show answer & explanation

Answer: $\frac{5}{2}$

Answer

The other solution to the equation is

\frac{5}{2}

.

The correct answer is found by using the relationship between the coefficients of a quadratic equation and the sum of its roots. For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

. Substituting the values

a = 2

and

b = -9

, we find that the sum of the roots is

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

. Since one of the solutions is given as

2

, the other solution must be

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

. Alternatively, one can find the constant

c

by substituting

x = 2

into the equation to get

2(2)^2 - 9(2) + c = 0

, which yields

c = 10

. Solving the resulting equation

2x^2 - 9x + 10 = 0

by factoring gives

(2x - 5)(x - 2) = 0

, which confirms the other solution is

\frac{5}{2}

.

Step-by-Step Solution

1

Identify the relationship between the coefficients and the sum of the roots for a quadratic equation.

For any quadratic equation in the form

ax^2 + bx + c = 0

, the sum of its roots,

x_1

and

x_2

, is given by the formula

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

.

This relationship allows us to find the second root directly without needing to calculate the value of the constant

c

first.

2

Substitute the given root and the coefficients of the quadratic equation into the sum of the roots formula.

Here,

a = 2

and

b = -9

. The given root is

x_1 = 2

. Substituting these values gives

2 + x_2 = -\frac{-9}{2}

, which simplifies to

2 + x_2 = \frac{9}{2}

.

This sets up a linear equation to solve for the unknown root,

x_2

.

3

Solve for the second root

x_2

.

x_2 = \frac{9}{2} - 2 = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}

.

This isolates the unknown variable to find the final value.

Key Concept

Using the sum of roots formula (

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

) to solve for an unknown solution of a quadratic equation.

Question 1767Question →

An artist creates custom ceramic tiles in two shapes: square tiles and hexagonal tiles. Each square tile has an area of $12$ square inches, and each hexagonal tile has an area of $28$ square inches. The artist is designing a mosaic with a total area of $460$ square inches using only these two types of tiles. If the artist uses $15$ square tiles, how many hexagonal tiles are used in the mosaic?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10. By setting up a linear equation representing the total area of the mosaic,

12s + 28h = 460

, and substituting the given value of 15 for the number of square tiles, we solve for the number of hexagonal tiles to get 10.

To find the number of hexagonal tiles used in the mosaic, we set up a linear equation in two variables representing the total area. Let

s

represent the number of square tiles, and let

h

represent the number of hexagonal tiles. The total area is the sum of the areas of the square tiles and the hexagonal tiles, which is

12s + 28h = 460

. Given that the artist uses

15

square tiles, we substitute

s = 15

into the equation to get

12(15) + 28h = 460

. Simplifying the equation yields

180 + 28h = 460

. Subtracting

180

from both sides gives

28h = 280

. Dividing both sides by

28

yields

h = 10

. Therefore, the artist uses 10 hexagonal tiles.

Step-by-Step Solution

1

Define variables and set up the linear equation representing the total area of the mosaic.

12s + 28h = 460

, where

s

is the number of square tiles and

h

is the number of hexagonal tiles.

This models the relationship between the quantities of each tile type used and the total surface area of the mosaic.

2

Substitute the given number of square tiles into the equation.

12(15) + 28h = 460

, which simplifies to

180 + 28h = 460

.

We are given that the artist uses exactly 15 square tiles in the design.

3

Isolate the variable term by subtracting the constant from both sides.

28h = 280

.

This isolates the term containing the unknown variable on one side of the equation.

4

Solve for the remaining variable by dividing both sides by its coefficient.

h = 10

.

Dividing both sides by the coefficient yields the final count of hexagonal tiles.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 1768Question →

In 2015, the population of a certain species of owl in a nature reserve was $x$ . By 2020, the population had increased by $35\%$ . Between 2020 and 2025, the population decreased by $20\%$ of its 2020 value. If the population of the owl species in 2025 was $270$ , what was the value of $x$ ?

Show answer & explanation

Answer: 250

Answer

The population in 2015 was 250.

To find the initial population

x

, we apply the percent changes sequentially. A

35\%

increase is represented by multiplying by

1.35

, and a

20\%

decrease is represented by multiplying by

0.80

. The final population in terms of

x

is

x \times 1.35 \times 0.80 = 1.08x

. Given that the population in 2025 is

270

, we set up the equation

1.08x = 270

. Dividing both sides by

1.08

yields

x = 250

.

Step-by-Step Solution

1

Write an expression for the owl population in 2020 in terms of

x

.

The population in 2020 was

1.35x

.

An increase of

35\%

is represented by multiplying the initial amount by

1 + 0.35 = 1.35

.

2

Write an expression for the owl population in 2025 in terms of

x

.

The population in 2025 was

1.08x

.

A decrease of

20\%

from the 2020 population is represented by multiplying the 2020 population by

1 - 0.20 = 0.80

. Thus, the 2025 population is

1.35x \times 0.80 = 1.08x

.

3

Set the expression for the 2025 population equal to the given value of 270 and solve for

x

.

x = 250

Solving the equation

1.08x = 270

by dividing both sides by

1.08

gives

x = \frac{270}{1.08} = 250

.

Key Concept

Consecutive percent changes

Estimated Time:1m 30s

Question 1769Question →

A company uses a fleet of refrigerated delivery trucks to transport fresh produce. The temperature inside a truck's cooling compartment, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), $t$ minutes after the cooling system is turned on is modeled by the equation below:

T = 68 - 1.25t

Where $0 \leq t \leq 24$ . Which of the following is the best interpretation of the number $68$ in this context?

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

The rate, in degrees Fahrenheit per minute, at which the temperature inside the compartment decreases

The temperature inside the compartment, in degrees Fahrenheit, 1 minute after the cooling system is turned on, which is

67.25^\circ\text{F}

The number of minutes it takes for the temperature inside the compartment to reach

0^\circ\text{F}

Show answer & explanation

Answer: The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

Answer

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

The correct answer correctly identifies the meaning of the constant term

68

as the y-intercept of the linear equation. In this context, the y-intercept represents the value of

T

when the time

t

is

0

, which corresponds to the initial temperature inside the compartment before any cooling occurs.

Step-by-Step Solution

1

Identify the form of the linear equation and its components.

The equation is in the slope-intercept form,

T = mt + b

, where

m = -1.25

is the slope and

b = 68

is the

T

-intercept (or vertical intercept).

Recognizing the mathematical meaning of each coefficient is the first step in interpreting them in a real-world context.

2

Determine the physical meaning of the vertical intercept (

b = 68

).

The vertical intercept occurs when the independent variable,

t

(time in minutes), is equal to

0

. Evaluating

T

at

t = 0

gives

T(0) = 68 - 1.25(0) = 68

.

Setting the independent variable to zero allows us to define the starting state of the system.

3

Interpret this starting state in the context of the problem.

At

t = 0

, the cooling system has just been turned on. Thus,

68^\circ\text{F}

represents the temperature inside the truck's cooling compartment when the system is first turned on.

Connecting the value of

T(0)

back to the scenario provides the correct interpretation of the constant

68

.

Key Concept

Interpreting the y-intercept of a linear model in context

Estimated Time:1m 30s

Question 1770Question →

A software update reduced the storage space used by an application on a smartphone by $30\%$ . If the application originally used $400$ megabytes ( $\text{MB}$ ) of storage space, how many megabytes of storage space does the application use after the update?

Show answer & explanation

Answer: 280

Answer

The application uses

280

megabytes of storage space after the update.

The correct answer is

280

. The reduction in storage space is

30\%

of the original

400

megabytes, which is calculated as

0.30 \times 400 = 120

megabytes. Subtracting this reduction from the original storage space gives

400 - 120 = 280

megabytes. Alternatively, a reduction of

30\%

means the remaining storage is

70\%

of the original storage, which is

0.70 \times 400 = 280

megabytes.

Step-by-Step Solution

1

Calculate the reduction in storage space.

The reduction is

120

megabytes.

Multiply the original storage space of

400

megabytes by the percent reduction expressed as a decimal (

0.30

).

2

Subtract the reduction from the original storage space.

The new storage space is

280

megabytes.

Subtracting the amount of space saved (

120

megabytes) from the original space (

400

megabytes) gives the remaining space used.

Key Concept

Calculating a decreased amount after a percentage reduction.

Question 1771Question →

If

x

is a real solution to the equation

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

what is the value of

5x - 4

?

6

-14

-5

5

Show answer & explanation

Answer: -5

Answer

-5

The correct answer is

-5

. Solving the equation by multiplying both sides by the least common denominator

(x-1)(x-2)

yields the quadratic equation

5x^2 - 9x - 2 = 0

. Factoring this equation gives the potential solutions

x = -1/5

and

x = 2

. However, substituting

x = 2

into the original equation results in division by zero, meaning

x = 2

is extraneous and must be discarded. The only valid solution is

x = -1/5

. Substituting this value into the expression

5x - 4

yields

5(-1/5) - 4 = -5

.

Step-by-Step Solution

1

Identify the least common denominator (LCD) and multiply both sides of the equation to clear the denominators.

The LCD is

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

. Multiplying both sides by the LCD gives:

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

Multiplying by the LCD simplifies the rational equation into a polynomial equation.

2

Expand the terms and collect like terms to form a quadratic equation.

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

5x^2 - 8x = x + 2

5x^2 - 9x - 2 = 0

This puts the equation in the standard quadratic form

ax^2 + bx + c = 0

.

3

Factor the quadratic equation to find potential solutions.

The quadratic factors as:

(5x + 1)(x - 2) = 0

This yields potential solutions

x = -\frac{1}{5}

and

x = 2

.

Factoring allows us to isolate the candidate values of

x

.

4

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

For

x = 2

, the denominators

x - 2

and

x^2 - 3x + 2

become zero, which is undefined. Therefore,

x = 2

is an extraneous solution. For

x = -\frac{1}{5}

, all denominators are non-zero, so it is the only valid solution.

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

5

Substitute the valid solution into the target expression

5x - 4

.

5\left(-\frac{1}{5}\right) - 4 = -1 - 4 = -5

Evaluating the expression with the only valid solution gives the final answer.

Key Concept

Solving rational equations by clearing denominators and checking for extraneous solutions.

Estimated Time:2m 30s

Question 1772Question →

The function $f$ is defined by $f(x) = x^2 - 6x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $f$ has vertex $A$ . The function $g$ is defined by $g(x) = -f(x - 6)$ , and its graph has vertex $B$ . If the distance between points $A$ and $B$ is $10$ , and $c > 10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 13

Answer

The value of

c

is

13

.

Completing the square for

f(x) = x^2 - 6x + c

gives

f(x) = (x - 3)^2 + c - 9

, which shows that vertex

A

is located at

(3, c - 9)

. The transformation

g(x) = -f(x - 6)

translates the graph

6

units to the right and reflects it vertically, giving vertex

B

the coordinates

(3 + 6, -(c - 9)) = (9, 9 - c)

. Using the distance formula, the distance between

A

and

B

is

\sqrt{(9 - 3)^2 + ((9 - c) - (c - 9))^2} = \sqrt{36 + (18 - 2c)^2}

. Setting this distance equal to

10

and squaring both sides gives

36 + (18 - 2c)^2 = 100

, which simplifies to

(18 - 2c)^2 = 64

. Taking the square root of both sides gives

18 - 2c = 8

or

18 - 2c = -8

, yielding solutions of

c = 5

or

c = 13

. Since the question specifies that

c > 10

, the correct value must be

13

.

Step-by-Step Solution

1

Rewrite the function

f(x) = x^2 - 6x + c

in vertex form by completing the square.

f(x) = (x - 3)^2 + c - 9

, which gives the coordinates of vertex

A

as

(3, c - 9)

.

Completing the square reveals the vertex

(h, k)

of a quadratic function in the form

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

.

2

Determine the vertex

B

of the graph of

g(x) = -f(x - 6)

by applying transformations to vertex

A(3, c - 9)

.

The horizontal shift of

f(x - 6)

moves the vertex to

(3 + 6, c - 9) = (9, c - 9)

. The reflection of

-f(x - 6)

negates the

y

-coordinate of the vertex, resulting in

B(9, 9 - c)

.

The transformation

f(x - h)

shifts a graph right by

h

units, and the transformation

-f(x)

reflects it across the

x

-axis.

3

Use the distance formula to set up an equation for the distance between

A(3, c - 9)

and

B(9, 9 - c)

.

\sqrt{(9 - 3)^2 + ((9 - c) - (c - 9))^2} = 10 \implies \sqrt{6^2 + (18 - 2c)^2} = 10

.

The distance

d

between points

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

4

Solve the radical equation

\sqrt{36 + (18 - 2c)^2} = 10

for

c

.

36 + (18 - 2c)^2 = 100 \implies (18 - 2c)^2 = 64 \implies 18 - 2c = 8

or

18 - 2c = -8

. This yields

c = 5

or

c = 13

.

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

5

Apply the constraint

c > 10

to choose the correct value for

c

.

c = 13

.

The question specifies that

c

must be greater than

10

, which excludes

c = 5

.

Key Concept

Using vertex form of quadratic equations to determine vertex coordinates and applying transformations (horizontal shifts and vertical reflections) to find key graphical points.

Question 1773Question →

Complete the passage below by typing the grammatically correct form of the noun 'horse' in the blank.

Fill in the blanks below

During the late nineteenth century, British photographer Eadweard Muybridge captured pioneering sequential images of animals in motion. By utilizing a complex series of custom-built cameras triggered by tripwires, Muybridge was able to freeze a running strides in a way that had never been seen before, conclusively proving that all four of the animal's hooves leave the ground simultaneously during a gallop.

Show answer & explanation

Answer

horse's

The sentence requires the singular possessive form of the noun 'horse' because the indefinite article 'a' specifies a single animal, and the noun 'strides' that follows requires a possessive modifier.

Step-by-Step Solution

1

Analyze the grammatical context surrounding the blank to determine if the noun should be singular or plural.

The blank is preceded by the singular indefinite article 'a' ('a running...'), which indicates that a singular noun is required.

The article 'a' must modify a singular noun phrase.

2

Determine if the noun needs to be possessive or non-possessive.

The blank is followed directly by the noun 'strides' ('...strides in a way...'), indicating that the noun in the blank must possess the strides. Therefore, a possessive form is required.

A noun modifying another noun to show ownership or association requires the possessive case.

3

Combine singular number and possessive case to form the correct word.

The singular possessive form of the noun 'horse' is 'horse's'.

Singular possessive nouns are formed by adding an apostrophe and an 's' to the singular noun.

Key Concept

Plural and Possessive Nouns and Pronouns

Estimated Time:1m 0s

Question 1774Question →

The mass of a radioactive sample, in grams, is modeled by the function $M(t) = M_0 \left(\frac{1}{8}\right)^{\frac{t}{15}}$ , where $M_0$ is the initial mass of the sample and $t$ is the time, in minutes, since the sample started decaying. Which of the following functions correctly models the mass of the sample, in grams, in terms of $s$ seconds after the sample started decaying?

M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{5}}

M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{2700}}

M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

M(s) = M_0 \left(\frac{1}{2}\right)^{12s}

Show answer & explanation

Answer: $M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}$

Answer

The correct function modeling the mass in terms of seconds is

M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

.

To express the mass in terms of

s

seconds, we substitute

t = \frac{s}{60}

into the original decay function because the time in minutes is equal to the number of seconds divided by 60. This gives

M(s) = M_0 \left(\frac{1}{8}\right)^{\frac{s}{60 \times 15}} = M_0 \left(\frac{1}{8}\right)^{\frac{s}{900}}

. Next, since the options have a base of

\frac{1}{2}

, we rewrite the base

\frac{1}{8}

as

\left(\frac{1}{2}\right)^3

. Applying the power of a power rule

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

, we multiply the exponent

\frac{s}{900}

by 3, which yields

M(s) = M_0 \left(\frac{1}{2}\right)^{3 \times \frac{s}{900}} = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

.

Step-by-Step Solution

1

Relate the time variables

t

(in minutes) and

s

(in seconds).

Since 1 minute is equivalent to 60 seconds, the relationship is

t = \frac{s}{60}

.

This substitution allows the function to take time input in seconds instead of minutes.

2

Substitute

t = \frac{s}{60}

into the exponent of the original model.

M(s) = M_0 \left(\frac{1}{8}\right)^{\frac{s/60}{15}} = M_0 \left(\frac{1}{8}\right)^{\frac{s}{900}}

This updates the function variable to seconds and simplifies the fractional exponent.

3

Rewrite the base

\frac{1}{8}

as a power of

\frac{1}{2}

and apply exponent rules.

Since

\frac{1}{8} = \left(\frac{1}{2}\right)^3

, we write

M(s) = M_0 \left(\left(\frac{1}{2}\right)^3\right)^{\frac{s}{900}} = M_0 \left(\frac{1}{2}\right)^{3 \times \frac{s}{900}} = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

.

Applying the power of a power rule,

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

, simplifies the function to its final form with a base of

\frac{1}{2}

.

Key Concept

Applying exponent rules to manipulate exponential bases and performing variable substitutions in contextual models.

Estimated Time:2m 0s

Question 1775Question →

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 1776Question →

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 1777Question →

Mary Cassatt's paintings of domestic life often captured the quiet, intimate interactions between mothers and children. In *The Child's Bath*, Cassatt's brushstrokes emphasize the mother's tender care, contrasting with the child's playful posture. The ______ unique perspective on maternal bonds challenged nineteenth-century artistic conventions, which typically favored idealized and distant representations of family life.

Which choice completes the text so that it conforms to the conventions of Standard English?

artist

artists

artist's

artists'

Show answer & explanation

Answer: artist's

Answer

artist's

The correct answer is 'artist's' because the passage refers specifically to the single painter Mary Cassatt, requiring a singular noun. Furthermore, since this noun modifies 'perspective', it must be possessive. The singular possessive form of 'artist' is created by adding an apostrophe followed by 's' ('artist's').

Step-by-Step Solution

1

Determine whether the target noun should be singular or plural.

The target noun refers to Mary Cassatt, who is a single individual described in the passage, so a singular noun is required.

Establishing the correct number (singular vs. plural) ensures the noun accurately reflects the subject of the context.

2

Determine whether the target noun should be possessive or non-possessive.

The noun modifies the noun 'perspective' to show ownership, indicating a possessive form is required.

Identifying the relationship between the target word and the noun it modifies determines whether an apostrophe is needed.

3

Combine the singular and possessive requirements to select the correct spelling.

The singular possessive form of 'artist' is 'artist's'.

Applying standard punctuation rules for singular possessive nouns involves adding an apostrophe followed by an 's'.

Key Concept

Plural and Possessive Nouns and Pronouns

Question 1778Question →

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 1779Question →

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 1780Question →

Given the system of equations below:

\begin{aligned} 3x - y &= 11 \\ 2x + 3y &= 11 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + 2y$ ?

4

6

8

9

Show answer & explanation

Answer: $6$

Answer

The value of

x + 2y

is

6

.

To solve the system, we can isolate

y

in the first equation to get

y = 3x - 11

. Substituting this into the second equation gives

2x + 3(3x - 11) = 11

, which simplifies to

11x - 33 = 11

, or

11x = 44

, resulting in

x = 4

. Substituting

x = 4

back into the equation for

y

gives

y = 3(4) - 11 = 1

. The value of the expression

x + 2y

is therefore

4 + 2(1) = 6

.

Step-by-Step Solution

1

Isolate

y

in the first equation.

y = 3x - 11

To prepare for substitution into the second equation.

2

Substitute the expression for

y

into the second equation and solve for

x

.

2x + 3(3x - 11) = 11 \implies 11x - 33 = 11 \implies 11x = 44 \implies x = 4

To find the value of

x

.

3

Substitute the value of

x

back into the isolated equation for

y

to solve for

y

.

y = 3(4) - 11 \implies y = 1

To find the value of

y

.

4

Evaluate the expression

x + 2y

using the values of

x

and

y

.

4 + 2(1) = 6

To find the final requested value.

Key Concept

Solving a system of linear equations using substitution or elimination, and evaluating a linear combination of the variables.

Alternative Method

Alternatively, the system can be solved using the elimination method. Multiply the first equation by

3

to align the

y

-coefficients:

9x - 3y = 33

. Add this equation to the second equation,

2x + 3y = 11

, to eliminate

y

, yielding

11x = 44

, so

x = 4

. Then substitute

x = 4

into either original equation to find

y = 1

.

Estimated Time:1m 30s

Answer

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