All practice questions

Question 1721Question →

A researcher uses the linear equation $T = 18.5 + 4.2d$ to estimate the temperature, $T$ , in degrees Celsius, of the Earth's crust at a depth of $d$ kilometers below the surface in a certain region. What is the estimated increase in temperature, in degrees Celsius, for each increase of 5 kilometers in depth?

Show answer & explanation

Answer: 21

Answer

21

The coefficient of

d

in the equation is

4.2

, representing a temperature increase of

4.2

degrees Celsius for every 1 kilometer increase in depth. To find the temperature increase for an increase of 5 kilometers in depth, multiply this rate of change by 5:

4.2 \times 5 = 21

.

Step-by-Step Solution

1

Identify the slope of the linear equation.

The slope is

4.2

.

In the equation

T = 18.5 + 4.2d

, the term

4.2d

indicates that for every 1 kilometer increase in depth (

d

), the temperature (

T

) increases by

4.2

degrees Celsius.

2

Calculate the total temperature increase for a 5-kilometer depth increase.

The increase is

21

degrees Celsius.

Since the rate of temperature increase is

4.2

degrees Celsius per kilometer, a depth increase of

5

kilometers results in a temperature increase of

4.2 \times 5 = 21

degrees Celsius.

Key Concept

Interpreting the slope of a linear equation in context as a constant rate of change.

Estimated Time:1m 30s

Question 1722Question →

What is the sum of all real solutions to the equation $\sqrt{3x + 7} - \frac{10}{\sqrt{3x + 7}} = 3$ ?

Show answer & explanation

Answer: 6

Answer

The only real solution is 6, so the sum of all real solutions is 6.

The correct answer is 6. By substituting

u = \sqrt{3x + 7}

, the equation becomes

u - 10/u = 3

, which simplifies to the quadratic

u^2 - 3u - 10 = 0

. Factoring gives

(u - 5)(u + 2) = 0

. Since the principal square root must be non-negative,

u

must be positive, so we reject

u = -2

and keep

u = 5

. Solving

\sqrt{3x + 7} = 5

by squaring both sides yields

3x + 7 = 25

, which simplifies to

x = 6

. The extraneous solution

x = -1

(which comes from

u = -2

) must be discarded because substituting it back into the original equation results in

-3

instead of

3

. Thus, the only real solution is 6.

Step-by-Step Solution

1

Substitute

u = \sqrt{3x + 7}

into the equation.

u - \frac{10}{u} = 3

, where

u > 0

.

This substitution simplifies the radical equation into a rational equation of a single variable.

2

Multiply the entire equation by

u

and solve the resulting quadratic equation.

u^2 - 3u - 10 = 0 \implies (u - 5)(u + 2) = 0

, which yields

u = 5

or

u = -2

.

Clearing the denominator transforms the rational expression into a standard quadratic equation.

3

Apply the constraint that

u = \sqrt{3x + 7}

must be non-negative.

u = 5

is the only valid root;

u = -2

has no real solution and must be discarded.

The output of a principal square root is non-negative by definition.

4

Substitute back

\sqrt{3x + 7} = 5

and solve for

x

.

3x + 7 = 25 \implies 3x = 18 \implies x = 6

.

Squaring both sides eliminates the radical and allows us to isolate the variable

x

.

5

Verify if solving

\sqrt{3x+7} = -2

leads to an extraneous solution.

Squaring both sides of

\sqrt{3x+7} = -2

gives

3x+7 = 4 \implies x = -1

. Substituting

x = -1

into the original equation yields

\sqrt{4} - \frac{10}{\sqrt{4}} = 2 - 5 = -3 \neq 3

. Therefore,

x = -1

is extraneous.

Verifying solutions is necessary to identify and discard values that do not satisfy the original equation.

Key Concept

Solving equations using substitution and identifying extraneous solutions

Estimated Time:2m 30s

Question 1723Question →

If $x$ is a real solution to the equation $\sqrt{2x + 7} - x = 2$ , what is the value of $x + 4$ ?

5

1

7

3

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. To solve the equation

\sqrt{2x + 7} - x = 2

, we first isolate the radical to get

\sqrt{2x + 7} = x + 2

. Squaring both sides yields

2x + 7 = x^2 + 4x + 4

. Rearranging this equation into standard quadratic form gives

x^2 + 2x - 3 = 0

, which factors as

(x + 3)(x - 1) = 0

. This gives two potential solutions:

x = 1

and

x = -3

. Checking these in the original equation shows that

x = 1

is valid, whereas

x = -3

is extraneous because

\sqrt{2(-3)+7} - (-3) = 1 + 3 = 4 \neq 2

. Therefore, the only real solution is

x = 1

. Substituting this into the expression

x + 4

gives

1 + 4 = 5

.

Step-by-Step Solution

1

Isolate the radical expression on one side of the equation.

\sqrt{2x + 7} = x + 2

This sets up the equation to eliminate the radical by squaring both sides.

2

Square both sides of the equation to eliminate the radical.

2x + 7 = (x + 2)^2

Squaring a square root removes the radical, allowing the equation to be solved algebraically.

3

Expand the squared binomial on the right side.

2x + 7 = x^2 + 4x + 4

Expanding

(x+2)^2

yields a quadratic term, a linear term, and a constant term.

4

Rearrange the equation to set it equal to zero.

x^2 + 2x - 3 = 0

Subtracting

2x

and

7

from both sides simplifies the equation into standard quadratic form.

5

Factor the quadratic equation.

(x + 3)(x - 1) = 0

Factoring allows us to find the potential solutions for

x

.

6

Identify potential solutions and check for extraneous solutions by substituting them back into the original equation.

For

x = 1

,

\sqrt{2(1) + 7} - 1 = 3 - 1 = 2

, which is true. For

x = -3

,

\sqrt{2(-3) + 7} - (-3) = 1 + 3 = 4

, which does not equal

2

.

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

x = -3

is extraneous, leaving

x = 1

as the only valid solution.

7

Calculate the value of the expression

x + 4

using the valid solution.

1 + 4 = 5

The question asks for the value of

x + 4

, so we substitute the valid solution

x = 1

into this expression.

Key Concept

Solving radical equations and checking for extraneous solutions.

Question 1724Question →

The graph of the polynomial function $f$ in the $xy$ -plane is defined by $f(x) = a(x - 3)(x + 4)$ , where $a$ is a constant. If the $y$ -intercept of the graph is $(0, -24)$ , what is the value of $a$ ?

-2

\frac{1}{2}

2

-12

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. The

y

-intercept of a graph is the point where

x = 0

. For the function

f(x) = a(x - 3)(x + 4)

, substituting

x = 0

yields

f(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a

. Given that the

y

-intercept is

(0, -24)

, the value of the function at

x = 0

is

-24

. Setting these two values equal gives the equation

-12a = -24

. Dividing both sides of the equation by

-12

isolates the constant, resulting in

a = 2

.

Step-by-Step Solution

1

Identify the relationship between the

y

-intercept and the function value at

x = 0

.

Since the

y

-intercept is

(0, -24)

, it follows that

f(0) = -24

.

By definition, the

y

-intercept of a graph is the point where the graph crosses the vertical axis, which occurs at

x = 0

.

2

Substitute

x = 0

into the polynomial expression for

f(x)

.

f(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a

.

This evaluates the algebraic expression at the

y

-intercept to express the value in terms of the constant

a

.

3

Equate the evaluated expression to the known

y

-intercept value and solve for

a

.

-12a = -24 \Rightarrow a = 2

.

Setting the two expressions for

f(0)

equal allows us to solve the linear equation for the constant

a

.

Key Concept

Evaluating a polynomial at

x = 0

to relate its algebraic form to its

y

-intercept.

Question 1725Question →

A commercial printing press uses a continuous roll of paper to print newspapers at a constant rate. After $10$ minutes of operation, the remaining length of the paper roll is $12,500$ feet. After $25$ minutes of operation, the remaining length of the paper roll is $8,000$ feet. If the relationship between the printing time, in minutes, and the remaining length of the paper roll, in feet, is linear, how many minutes after the printing press starts operating will the remaining length of the paper roll be $2,000$ feet?

Show answer & explanation

Answer: 45

Answer

The remaining length of the paper roll will be

2,000

feet after

45

minutes of operation.

We are given that the remaining length of the paper roll is a linear function of time,

t

. Let

L(t)

be the remaining length of the paper roll, in feet, after

t

minutes of operation. We can represent the given information as two coordinate points:

(10, 12500)

and

(25, 8000)

. First, find the slope, which represents the constant rate at which the paper is consumed:

m = \frac{8000 - 12500}{25 - 10} = \frac{-4500}{15} = -300

feet per minute. Next, write the linear equation using the point-slope form:

L(t) - 12500 = -300(t - 10)

, which simplifies to

L(t) = 15500 - 300t

. To find the time when the remaining length is

2,000

feet, set

L(t) = 2000

and solve for

t

:

2000 = 15500 - 300t

, which simplifies to

300t = 13500

, giving

t = 45

.

Step-by-Step Solution

1

Calculate the rate of paper consumption (the slope of the linear equation) using the two given points,

(10, 12500)

and

(25, 8000)

.

The rate of paper consumption is

-300

feet per minute.

To establish the linear relationship, we first need the constant rate of change (slope) from the two known coordinate points.

2

Use the point-slope equation of a line,

y - y_1 = m(x - x_1)

, with the point

(10, 12500)

and slope

m = -300

, to find the equation relating the remaining length,

L

, to the time,

t

.

L(t) = 15500 - 300t

We need the full linear model to calculate the remaining length at any specific time.

3

Substitute

L(t) = 2000

into the linear equation and solve for

t

.

t = 45

This gives the specific operating time in minutes when the remaining paper roll length is

2,000

feet.

Key Concept

Writing and solving linear equations in two variables from two coordinate points.

Question 1726Question →

What is the only real solution to the equation $\sqrt{3x + 13} = x + 3$ ?

-4

Both

-4

and

1

1

-1

Show answer & explanation

Answer: $1$

Answer

The correct answer is the value 1.

Squaring both sides of the equation

\sqrt{3x + 13} = x + 3

yields

3x + 13 = x^2 + 6x + 9

. Rearranging this into standard quadratic form gives

x^2 + 3x - 4 = 0

. Factoring the quadratic expression gives

(x + 4)(x - 1) = 0

, which yields the potential solutions

x = 1

and

x = -4

. Substituting

x = 1

back into the original equation gives

\sqrt{16} = 4

, which is true. Therefore, the value

1

is the only real solution.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

3x + 13 = (x + 3)^2

Squaring both sides eliminates the radical, allowing us to solve for

x

.

2

Expand the right side and rewrite the equation in standard quadratic form.

x^2 + 3x - 4 = 0

Expanding

(x+3)^2

yields

x^2 + 6x + 9

. Subtracting

3x

and

13

from both sides sets the quadratic equation to zero.

3

Factor the quadratic equation.

(x + 4)(x - 1) = 0

We find two numbers that multiply to

-4

and add to

3

, which are

4

and

-1

.

4

Solve for the potential solutions.

x = -4

or

x = 1

Setting each factor to zero yields the potential solutions.

5

Verify each potential solution in the original equation.

x = 1

is valid, and

x = -4

is extraneous.

Plugging in

x = 1

gives

\sqrt{16} = 4

, which is true. Plugging in

x = -4

gives

\sqrt{1} = -1

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

Key Concept

Solving radical equations and checking for extraneous solutions.

Estimated Time:45s

Question 1727Question →

In the field of paleontology, the preservation of soft-bodied organisms is exceedingly rare because organic decay typically destroys tissues before fossilization can occur. Recently, a team of researchers discovered exceptionally preserved soft-tissue fossils in the Burgess Shale ______ these specimens provide rare insights into the internal anatomy of early Cambrian organisms that lacked hard shells. Which choice completes the text so that it conforms to the conventions of Standard English?

Shale;

Shale,

Shale

Shale, despite

Show answer & explanation

Answer: Shale;

Answer

The correct answer is the option containing the semicolon, which properly separates the two independent clauses.

The correct answer is the option ending with a semicolon. A semicolon is a grammatically valid way to connect two independent clauses without any coordinating conjunctions.

Step-by-Step Solution

1

Analyze the structure of the clauses flanking the blank.

The clause before the blank ('Recently, a team of researchers discovered exceptionally preserved soft-tissue fossils in the Burgess Shale') and the clause after the blank ('these specimens provide rare insights into the internal anatomy of early Cambrian organisms that lacked hard shells') are both independent clauses.

To determine how they must be grammatically linked.

2

Determine the punctuation rules for joining independent clauses.

Independent clauses must be joined by a period, a semicolon, a colon, or a comma with a coordinating conjunction.

To avoid run-on sentences and comma splices.

3

Evaluate the choices against these rules.

Only the semicolon option correctly connects the two independent clauses without introducing a grammatical error.

To identify the correct answer choice.

Key Concept

Connecting independent clauses using semicolons to avoid run-on sentences and comma splices.

Question 1728Question →

A software developer uses the equation $P = 1500 + 45n$ to calculate the total price $P$ , in dollars, charged to a client for a project, where $n$ represents the number of hours spent debugging the software. Which of the following is the best interpretation of the number $45$ in this context?

The increase in the total price, in dollars, for each additional hour spent debugging the software

The initial price, in dollars, charged for the project before any hours are spent debugging

The number of additional hours spent debugging for each 1-dollar increase in the total price

The total price, in dollars, charged for a project that requires exactly 1 hour of debugging

Show answer & explanation

Answer: The increase in the total price, in dollars, for each additional hour spent debugging the software

Answer

The increase in the total price, in dollars, for each additional hour spent debugging the software

The coefficient of

n

in the equation

P = 1500 + 45n

is

45

, which represents the slope of the line. The slope measures the rate of change of the dependent variable,

P

(total price in dollars), per unit change in the independent variable,

n

(hours spent debugging). Thus,

45

represents an increase of 45 dollars in the total price for each additional hour spent debugging.

Step-by-Step Solution

1

Identify the structure of the linear equation.

The equation

P = 1500 + 45n

is in the slope-intercept form

y = mx + b

, where

P

is the dependent variable (equivalent to

y

),

n

is the independent variable (equivalent to

x

), the coefficient

45

is the slope

m

, and the constant

1500

is the y-intercept

b

.

Understanding the components of a linear equation helps map them to their real-world meanings.

2

Interpret the meaning of the slope in context.

The slope

m = 45

represents the rate of change of the dependent variable

P

(total price in dollars) with respect to the independent variable

n

(hours spent debugging). This means that for each unit increase in

n

(1 additional hour of debugging),

P

increases by

45

units (45 dollars).

The coefficient of the independent variable in a linear model represents the unit rate of change.

Key Concept

Interpreting the slope of a linear model in context

Question 1729Question →

If $x$ and $y$ satisfy the system of equations below, what is the value of $x + y$ ?

\begin{aligned} 8^{x-y} &= 2^{y+3} \\ 9^{x} &= 27^{y-1} \end{aligned}

Show answer & explanation

Answer: 36

Answer

36

The correct answer is 36. By rewriting

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

with base 2, we obtain the relation

3x - 4y = 3

. Similarly, rewriting

9^x = 27^{y-1}

with base 3 gives

2x - 3y = -3

. Solving this system of equations simultaneously yields

x = 21

and

y = 15

. Adding these values together gives the final result of 36.

Step-by-Step Solution

1

Rewrite the first equation using the common base of 2.

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

, which simplifies to

3x - 4y = 3

.

Since

8 = 2^3

, we can apply the power of a power rule

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

to rewrite

8^{x-y}

as

2^{3(x-y)}

. Setting the exponents equal gives the linear equation

3(x-y) = y+3

, which simplifies to

3x - 4y = 3

.

2

Rewrite the second equation using the common base of 3.

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

, which simplifies to

2x - 3y = -3

.

Since

9 = 3^2

and

27 = 3^3

, we rewrite the terms using base 3. Equating the exponents gives

2x = 3(y-1)

, which simplifies to

2x - 3y = -3

.

3

Solve the system of two linear equations.

x = 21

and

y = 15

.

Using the elimination method, we can multiply the first equation by 3 and the second equation by 4 to align the coefficients of

y

, obtaining

9x - 12y = 9

and

8x - 12y = -12

. Subtracting the equations gives

x = 21

. Substituting this back yields

y = 15

.

4

Find the sum of

x

and

y

.

36

The question asks for the value of

x + y

. Substituting

x = 21

and

y = 15

gives

21 + 15 = 36

.

Key Concept

Solving systems of exponential equations by expressing bases as powers of common bases to establish a system of linear equations.

Question 1730Question →

If $\sqrt{4x + 9} = 5$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

Squaring both sides of the equation

\sqrt{4x + 9} = 5

yields

4x + 9 = 25

. Subtracting 9 from both sides gives

4x = 16

, and dividing by 4 yields

x = 4

. Checking this solution in the original equation yields

\sqrt{4(4) + 9} = \sqrt{25} = 5

, which confirms it is correct.

Step-by-Step Solution

1

Square both sides of the equation

4x + 9 = 25

To eliminate the square root on the left side of the equation.

2

Subtract 9 from both sides of the equation

4x = 16

To isolate the variable term on the left side of the equation.

3

Divide both sides of the equation by 4

x = 4

To solve for

x

.

4

Substitute the solved value back into the original equation

\sqrt{4(4) + 9} = 5 \implies 5 = 5

To ensure the solution is valid and not extraneous.

Key Concept

Solving radical equations of the form

\sqrt{ax + b} = c

by squaring both sides.

Question 1731Question →

A line in the $xy$ -plane representing the linear function $f$ has a $y$ -intercept of $(0, 8)$ and contains the point $(a, b)$ , where $a$ and $b$ are positive integers. If the slope of this line is $-\frac{3}{5}$ and the sum of $a$ and $b$ is $12$ , what is the value of $a$ ?

2

6

8

10

Show answer & explanation

Answer: 10

Answer

10

Since the line representing the function has a y-intercept of (0, 8) and a slope of -3/5, its equation in slope-intercept form is y = -3/5x + 8. Since the line contains the point (a, b), substituting x = a and y = b into this equation gives b = -3/5a + 8. The sum of a and b is given as 12, so a + b = 12, which can be rewritten as b = 12 - a. Substituting this expression for b gives the equation 12 - a = -3/5a + 8. Subtracting 8 from both sides yields 4 - a = -3/5a. Adding a to both sides yields 4 = 2/5a. Multiplying both sides by 5/2 gives a = 10.

Step-by-Step Solution

1

Write the equation of the linear function in slope-intercept form.

y = -\frac{3}{5}x + 8

The y-intercept of the line is given as (0, 8), and the slope of the line is given as -3/5.

2

Substitute the point (a, b) into the equation of the line.

b = -\frac{3}{5}a + 8

Since the point (a, b) lies on the line, its coordinates must satisfy the equation of the line.

3

Express b in terms of a using the given sum relationship.

b = 12 - a

We are given that the sum of a and b is 12, which can be written as a + b = 12.

4

Substitute the expression for b into the line's equation and solve for a.

a = 10

Substituting b = 12 - a into b = -3/5a + 8 yields 12 - a = -3/5a + 8, which simplifies to 4 = 2/5a, leading to a = 10.

Key Concept

Linear Functions and Graphs

Question 1732Question →

The sociable weaver bird (*Philetairus socius*) of southern Africa constructs massive communal nests that can house hundreds of individuals across multiple generations. While these structures offer excellent protection from predators and extreme weather, a single __________ weight can eventually become so great that it snaps the large tree branch supporting the entire colony.

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

nest's

nest

nests

nests'

Show answer & explanation

Answer: nest's

Answer

nest's

The correct answer is the singular possessive noun 'nest's'. The text refers to 'a single' structure, which requires the singular form of the noun. Because the weight belongs to the nest, the possessive form with an apostrophe followed by 's' is grammatically correct.

Step-by-Step Solution

1

Analyze the context around the blank to determine whether the target noun should be singular or plural.

The phrase 'a single' preceding the blank indicates that a singular noun is required.

Determining grammatical number is essential for selecting the correct noun form.

2

Determine if the noun needs to be possessive or non-possessive based on its syntactic relationship with the following word.

The noun directly precedes 'weight', which is a noun it modifies. The weight belongs to the nest, so the possessive form is required.

Identifying the relationship of possession helps place the apostrophe correctly.

3

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

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

This matches standard English conventions for possessive singular nouns.

Key Concept

Plural and Possessive Nouns

Question 1733Question →

Read the passage. Which punctuation mark (colon, semicolon, or dash) should be placed in the blank to properly connect the clauses according to the conventions of Standard English?

Fill in the blanks below

Although celestial displays of light can occur at both poles, they are distinct geographical phenomena. In the high-latitude regions of the Northern Hemisphere, the colorful display known as the aurora borealis is caused by the collision of charged solar particles with gases in Earth's atmosphere in the Southern Hemisphere, a nearly identical phenomenon occurs under the name aurora australis.

Show answer & explanation

Answer

A semicolon (;) is the correct punctuation mark to connect the two independent clauses.

A semicolon (;) is the correct punctuation mark to connect the two independent clauses. The clause preceding the blank ('In the high-latitude regions of the Northern Hemisphere, the colorful display known as the aurora borealis is caused by the collision of charged solar particles with gases in Earth's atmosphere') and the clause following the blank ('in the Southern Hemisphere, a nearly identical phenomenon occurs under the name aurora australis') are both independent. Because they are closely related and not joined by a coordinating conjunction (like 'and' or 'but'), a semicolon is required to connect them grammatically.

Step-by-Step Solution

1

Analyze the clause before the blank.

The clause 'In the high-latitude regions of the Northern Hemisphere, the colorful display known as the aurora borealis is caused by the collision of charged solar particles with gases in Earth's atmosphere' is independent because it contains a subject and a verb and can stand alone as a complete sentence.

To determine correct punctuation, we must first establish whether the clauses are independent or dependent.

2

Analyze the clause after the blank.

The clause 'in the Southern Hemisphere, a nearly identical phenomenon occurs under the name aurora australis' is also independent because it contains a subject and a verb and can stand alone as a complete sentence.

Identifying the grammatical nature of the second clause tells us how they can be legally joined.

3

Select the appropriate punctuation mark to link the two independent clauses.

Since both clauses are independent and closely related, and there is no coordinating conjunction (such as 'and' or 'but'), a semicolon is the correct punctuation mark to join them without creating a run-on sentence or a comma splice.

Standard English conventions require a semicolon, a colon, or a period to join two independent clauses when no coordinating conjunction is present. A semicolon is appropriate here because the second clause provides a closely related, parallel point.

Key Concept

Using a semicolon to link two independent clauses

Question 1734Question →

The polynomial function $p$ is defined by $p(x) = (x - 3)(x - 5)(x - r)$ , where $r$ is a constant. If the graph of $y = p(x)$ in the $xy$ -plane intersects the $y$ -axis at $(0, -60)$ , what is the value of $r$ ?

Show answer & explanation

Answer: 4

Answer

The value of

r

is

4

.

Evaluating the polynomial at

x = 0

yields the

y

-intercept of the graph, which is given as

-60

. Substituting

0

for

x

in the equation

p(x) = (x - 3)(x - 5)(x - r)

gives

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

. Setting this expression equal to

-60

yields

-15r = -60

. Dividing both sides of the equation by

-15

yields

r = 4

.

Step-by-Step Solution

1

Identify the meaning of the

y

-intercept.

Evaluating

p(x)

at

x = 0

must equal

-60

, so

p(0) = -60

.

The

y

-intercept of a graph occurs where the

x

-coordinate is

0

.

2

Substitute

x = 0

into the polynomial expression.

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

.

This simplifies the polynomial expression in terms of the unknown constant

r

.

3

Set the expression equal to the given

y

-intercept value and solve for

r

.

-15r = -60 \implies r = 4

.

Dividing both sides by

-15

isolates the variable

r

.

Key Concept

Using the y-intercept of a polynomial function to find an unknown root/constant in its factored form.

Question 1735Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has a vertex $(h, k)$ that lies on the line $y = 4x + 8$ . If the graph of $y = f(x)$ has $x$ -intercepts at $x = -2$ and $x = 6$ , what is the value of $c$ ?

Show answer & explanation

Answer: 12

Answer

The value of c is 12.

The correct answer is 12. The axis of symmetry of the quadratic function lies midway between its x-intercepts at

x = -2

and

x = 6

, giving an x-coordinate of

h = 2

for the vertex. Since the vertex lies on the line

y = 4x + 8

, its y-coordinate is

k = 4(2) + 8 = 16

. Substituting the vertex

(2, 16)

into the vertex form of a quadratic function gives

f(x) = a(x - 2)^2 + 16

. Using the x-intercept at

(6, 0)

to solve for

a

gives

0 = a(6 - 2)^2 + 16

, which yields

a = -1

. Evaluating the function at

x = 0

to find the constant term

c

gives

c = f(0) = -(0 - 2)^2 + 16 = 12

.

Step-by-Step Solution

1

Determine the x-coordinate of the vertex of the quadratic function.

h = 2

The axis of symmetry of a parabola lies midway between its x-intercepts. The midpoint of the intercepts

x = -2

and

x = 6

is calculated as

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

.

2

Determine the y-coordinate of the vertex using the given linear equation.

k = 16

Since the vertex

(h, k)

lies on the line

y = 4x + 8

, substituting

x = h = 2

into the line equation gives

k = 4(2) + 8 = 16

.

3

Write the quadratic function in vertex form and solve for the leading coefficient

a

.

a = -1

The vertex form is

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

. Substituting the vertex

(2, 16)

yields

f(x) = a(x - 2)^2 + 16

. Using the x-intercept

(6, 0)

, substitute

x = 6

and

f(x) = 0

into the equation:

0 = a(6 - 2)^2 + 16

, which simplifies to

16a = -16

, resulting in

a = -1

.

4

Expand the quadratic function or evaluate it at

x = 0

to find the value of

c

.

c = 12

In standard form

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

, the constant term

c

is equal to

f(0)

. Substituting

x = 0

into

f(x) = -(x - 2)^2 + 16

yields

f(0) = -(0 - 2)^2 + 16 = -4 + 16 = 12

.

Key Concept

Using symmetry and the vertex form of a quadratic function to determine its standard form coefficients.

Alternative Method

Alternatively, since the x-intercepts are

-2

and

6

, the quadratic function can be written in factored form as

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

. Expanding this gives

f(x) = a(x^2 - 4x - 12) = ax^2 - 4ax - 12a

. Comparing this to the standard form

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

, we see that the x-coordinate of the vertex is

h = -\frac{b}{2a} = -\frac{-4a}{2a} = 2

. Using the line equation

y = 4x + 8

at

x = 2

, we find the vertex y-coordinate is

k = 16

. Since the vertex is

(2, 16)

, we evaluate the factored form at

x = 2

:

16 = a(2 + 2)(2 - 6) \Rightarrow 16 = -16a \Rightarrow a = -1

. The constant term

c

is

-12a

, so

c = -12(-1) = 12

.

Estimated Time:2m 30s

Question 1736Question →

The price of a book was increased by $20\%$ to $\$30$ . What was the original price, in dollars, of the book?

24

25

36

6

Show answer & explanation

Answer: 25

Answer

The original price of the book was $25.

The correct answer is the price of

25. Let

x

represent the original price. Since the price was increased by

20\%

, the new price is

120\%

of

x

, or

1.20x

. Setting up the equation

1.20x = 30

and dividing both sides by

1.20

gives

x = 25$.

Step-by-Step Solution

1

Set up an equation representing the percent increase. Let

x

represent the original price of the book. A

20\%

increase means the final price is

120\%

of the original price.

1.20x = 30

To find the original price, we must relate the original price, the percent increase, and the final price.

2

Solve for

x

by dividing both sides of the equation by

1.20

.

x = \frac{30}{1.20} = 25

Dividing isolates the variable

x

to find the original price.

Key Concept

Solving for the initial value after a percentage increase.

Question 1737Question →

If $(x, y)$ is a solution to the system of equations below and $x > 1$ , what is the value of $x$ ?

\begin{aligned} y - 2x &= 3 \\ y^2 - 5x^2 &= 20 \end{aligned}

Show answer & explanation

Answer: 11

Answer

The value of

x

is 11.

The system can be solved by substituting the first equation into the second. Rearranging the first equation gives

y = 2x + 3

. Substituting this expression for

y

in the second equation gives

(2x + 3)^2 - 5x^2 = 20

. Expanding the binomial yields

4x^2 + 12x + 9 - 5x^2 = 20

, which simplifies to

-x^2 + 12x + 9 = 20

. Rearranging into standard quadratic form gives

x^2 - 12x + 11 = 0

. Factoring the quadratic equation results in

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

. Thus,

x = 11

or

x = 1

. Since the problem specifies

x > 1

, the value of

x

must be 11.

Step-by-Step Solution

1

Rearrange the first equation to express

y

in terms of

x

.

y = 2x + 3

This allows for substitution into the second equation.

2

Substitute

y = 2x + 3

into the second equation.

(2x + 3)^2 - 5x^2 = 20

To eliminate

y

and obtain an equation in terms of

x

only.

3

Expand and simplify the equation.

x^2 - 12x + 11 = 0

Expanding

(2x + 3)^2

gives

4x^2 + 12x + 9

. Subtracting

5x^2

yields

-x^2 + 12x + 9 = 20

. Subtracting 20 from both sides gives

-x^2 + 12x - 11 = 0

. Multiplying the entire equation by

-1

results in standard form.

4

Factor the quadratic equation.

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

To find the possible values of

x

.

5

Apply the given constraint

x > 1

.

x = 11

The solutions are

x = 11

and

x = 1

. The constraint

x > 1

excludes

x = 1

, leaving

x = 11

as the only valid solution.

Key Concept

Solving a system of nonlinear equations by substitution and factoring a quadratic equation.

Question 1738Question →

An online retailer offers two shipping options for heavy packages. The first option charges a flat fee of $15.00$ dollars plus $1.50$ dollars per pound of the package's weight. The second option charges a flat fee of $27.00$ dollars plus $0.70$ dollars per pound of the package's weight. If a package of weight $w$ pounds costs the same total amount of $c$ dollars under both options, what is the value of $w$ ?

1.5

37.5

52.5

15

Show answer & explanation

Answer: 15

Answer

15

To find the weight of the package that yields the same cost under both shipping options, we set the cost expressions equal to each other. The first option is represented by

c = 1.50w + 15

and the second by

c = 0.70w + 27

. Setting them equal gives

1.50w + 15 = 0.70w + 27

. Subtracting

0.70w

from both sides results in

0.80w + 15 = 27

. Subtracting

15

from both sides gives

0.80w = 12

. Dividing by

0.80

yields

w = 15

. Therefore, the weight of the package is 15 pounds.

Step-by-Step Solution

1

Set up the linear equations representing the total cost c for each shipping option based on the package's weight w.

The first option is represented by

c = 1.50w + 15

, and the second option is represented by

c = 0.70w + 27

.

This models the real-world scenario into a system of two linear equations with two variables.

2

Set the two expressions for the total cost c equal to each other to solve for the weight w.

1.50w + 15 = 0.70w + 27

Since the package costs the same under both options, the cost values are equal at this specific weight.

3

Isolate the variable w by subtracting

0.70w

and

15

from both sides of the equation.

0.80w = 12

, which simplifies to

w = 15

.

Subtracting

0.70w

from both sides isolates the variable terms on the left, and subtracting

15

isolates the constants on the right. Dividing by

0.80

solves for w.

Key Concept

Solving a system of linear equations by setting the expressions equal to find the point where two rates intersect.

Estimated Time:1m 30s

Question 1739Question →

If $x$ is a real number that satisfies the equation $\frac{x}{x-3} - \frac{6}{x^2-3x} = \frac{2}{x}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. Multiplying the entire equation by the lowest common denominator,

x(x-3)

, yields

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

. Distributing the right side gives

x^2 - 6 = 2x - 6

. Adding 6 to both sides and subtracting

2x

results in

x^2 - 2x = 0

. Factoring this equation gives

x(x-2) = 0

, which has solutions

x = 0

and

x = 2

. However,

x = 0

is extraneous because it results in division by zero in the original equation. Thus, the only valid real solution is 2.

Step-by-Step Solution

1

Identify the common denominator and multiply the equation by it.

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

To clear the fractions and simplify the rational equation.

2

Distribute and simplify the equation.

x^2 - 2x = 0

To set up the quadratic equation in standard form.

3

Factor and solve for

x

.

x = 0

or

x = 2

To find potential solutions to the equation.

4

Check for extraneous solutions by substituting the values back into the original denominators.

x = 2

x = 0

makes the denominators equal to zero and is therefore extraneous.

Key Concept

Solving rational equations and identifying extraneous solutions.

Question 1740Question →

If $\frac{x - 3}{4} = \frac{9}{x - 3}$ and $x > 3$ , what is the value of $x$ ?

Show answer & explanation

Answer: 9

Answer

9

Cross-multiplying the equation

\frac{x - 3}{4} = \frac{9}{x - 3}

gives

(x - 3)^2 = 36

. Taking the square root of both sides gives

x - 3 = 6

or

x - 3 = -6

. Solving for

x

yields

x = 9

or

x = -3

. Since the question specifies the constraint

x > 3

, the only valid solution is

9

.

Step-by-Step Solution

1

Cross-multiply the equation to eliminate the denominators.

(x - 3)^2 = 36

Multiplying both sides of the equation by

4(x - 3)

simplifies the rational equation into a quadratic form.

2

Take the square root of both sides of the equation.

x - 3 = 6

or

x - 3 = -6

Taking the square root of a squared term yields both positive and negative root options.

3

Solve each linear equation for

x

and apply the constraint

x > 3

.

x = 9

Adding

3

to both sides gives

x = 9

or

x = -3

. The constraint

x > 3

excludes the negative solution, leaving

9

as the only valid value.

Key Concept

Solving rational equations by cross-multiplication and factoring the resulting quadratic equation while adhering to domain constraints.

Answer

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