All practice questions

Question 341Question →

A photographer is buying a new camera bag. The original price of the bag is $\$80$ . If the store is having a clearance sale and offers a $25\%$ discount, what is the final sale price of the camera bag in dollars?

Show answer & explanation

Answer: 60

Answer

The final sale price of the camera bag is

\$60

.

To find the final sale price, first calculate the discount amount by multiplying the original price (

\$80

) by the discount rate (

25\%

or

0.25

), which equals

\$20

. Then, subtract this discount from the original price (

\$80 - \$20

) to get the final sale price of

\$60

.

Step-by-Step Solution

1

Calculate the discount amount in dollars.

\$20

The discount is

25\%

of the original price, so we multiply

\$80

by

0.25

.

2

Calculate the final sale price by subtracting the discount from the original price.

\$60

The final sale price is the original price minus the discount savings (

\$80 - \$20

).

Key Concept

Calculating a single-step discount and final sale price.

Question 342Question →

An armored battalion maintains a strict ratio of combat crew members to maintenance personnel of $7:3$ . If there are $210$ combat crew members currently assigned to the battalion, how many maintenance personnel are assigned?

300

63

490

90

Show answer & explanation

Answer: $90$

Answer

There are

90

maintenance personnel assigned to the battalion.

The ratio of combat crew to maintenance personnel is given as

7:3

. This means for every

7

combat crew members, there are

3

maintenance personnel. Since there are

210

combat crew members, we can set up the proportion:

7/3 = 210/M

. Solving for

M

, we get

7M = 630

, which means

M = 90

.

Step-by-Step Solution

1

Set up a proportion using the given ratio of combat crew to maintenance personnel.

\frac{7}{3} = \frac{210}{M}

Proportions allow us to find a missing quantity when the ratio between two parts is known.

2

Solve the proportion for the number of maintenance personnel (

M

).

7 \times M = 210 \times 3 \Rightarrow 7M = 630 \Rightarrow M = 90

Cross-multiplying and then dividing by

7

isolates the variable

M

to provide the final answer.

Key Concept

Solving for a missing part in a two-term ratio

Estimated Time:1m 0s

Question 343Question →

A naval submarine travels completely submerged at a steady cruising speed of $40\text{ knots}$ . If it maintains this speed for $3\text{ hours and } 15\text{ minutes}$ during a continuous sonar patrol, what is the total distance covered?

120\text{ nautical miles}

126\text{ nautical miles}

130\text{ nautical miles}

720\text{ nautical miles}

Show answer & explanation

Answer: $130\text{ nautical miles}$

Answer

130\text{ nautical miles}

The correct answer is derived by converting the

15\text{ minutes}

into

0.25\text{ hours}

, yielding a total continuous travel time of

3.25\text{ hours}

. Multiplying the cruising speed of

40\text{ knots}

by

3.25\text{ hours}

results in exactly

130\text{ nautical miles}

.

Step-by-Step Solution

1

Identify the standard distance formula.

\text{Distance} = \text{Rate} \times \text{Time}

This formula connects speed, travel time, and distance covered.

2

Convert the mixed time measurement (

3\text{ hours and } 15\text{ minutes}

) entirely into hours.

15\text{ minutes} \div 60\text{ minutes/hour} = 0.25\text{ hours}

. Total time is

3 + 0.25 = 3.25\text{ hours}

.

The rate is given in knots (nautical miles per hour), so the time must be strictly in hours before multiplication.

3

Multiply the rate by the total time in hours.

40 \times 3.25 = 130\text{ nautical miles}

Applying the values to the distance formula yields the total traveled distance.

Key Concept

Calculating distance by standardizing mixed time units (hours and minutes) into a single metric before multiplying by rate.

Question 344Question →

An entrepreneur purchases $5$ used electric scooters for $\$120$ each. She pays an additional flat fee of $\$150$ to replace the batteries for the entire batch. If she sells all $5$ scooters for $\$210$ each, what is her overall profit percentage on this investment?

Show answer & explanation

Answer: 40

Answer

40

The correct answer is found by first calculating the total investment, which includes both the purchase price of the scooters (

600) and the battery replacement costs (

150), yielding a total cost basis of

750. The total revenue from sales is

1,050. Subtracting the total cost from the revenue gives a net profit of

300. Finally, dividing the

300 profit by the $750 total investment yields a profit percentage of 40%.

Step-by-Step Solution

1

Calculate the total cost of purchasing the used scooters.

5 \times \$120 = \$600

This establishes the initial base cost before overhead.

2

Calculate the total investment by adding the battery replacement cost.

\$600 + \$150 = \$750

Profit must be calculated using the total cost basis, including all overhead expenses.

3

Calculate the total revenue from selling the scooters.

5 \times \$210 = \$1,050

Total revenue is needed to find the net profit.

4

Determine the net profit.

\$1,050 - \$750 = \$300

Net profit is total revenue minus total investment.

5

Calculate the profit percentage.

\frac{300}{750} \times 100 = 40\%

Profit percentage is the net profit divided by the total investment (cost basis), expressed as a percentage.

Key Concept

Calculating profit percentage by factoring in both unit costs and multi-step overhead expenses.

Estimated Time:1m 0s

Question 345Question →

A military cargo plane needs to transport supplies to a base that is $900$ miles away. If the plane flies at a constant speed of $300$ miles per hour, how many hours will the flight take?

Show answer & explanation

Answer: 3

Answer

The flight will take 3 hours.

To find the travel time, the total distance must be divided by the constant rate of speed. By applying the formula

\text{Time} = \frac{\text{Distance}}{\text{Rate}}

, dividing the

900

mile distance by the

300

mile per hour speed yields exactly

3

hours.

Step-by-Step Solution

1

Identify the known values from the word problem.

\text{Distance} = 900

miles,

\text{Rate} = 300

miles per hour.

These are the two variables required to solve for the missing variable, which is time.

2

Apply the time formula.

\text{Time} = \frac{900}{300} = 3

.

Time is equal to total distance divided by the constant rate.

Key Concept

Calculating elapsed time using the distance and rate formula.

Estimated Time:45s

Question 346Question →

During the mathematical modeling of an electromagnetic field, an engineer derives the formula

\frac{(4a^3b^{-2})^{-2}}{2a^{-4}b^5}

to represent the field intensity. Which of the following expressions is mathematically equivalent to this derived formula?

\frac{2}{a^2b}

\frac{a^5}{32b^9}

\frac{1}{32a^2b}

\frac{1}{32a^{10}b}

Show answer & explanation

Answer: $\frac{1}{32a^2b}$

Answer

The mathematically equivalent expression is

\frac{1}{32a^2b}

.

To find the equivalent expression, first apply the power of a power rule to the numerator. Distribute the exponent -2 to each factor: 4^(-2) * (a^3)^(-2) * (b^-2)^(-2), which simplifies to (1/16) * a^-6 * b^4. Next, divide this result by the denominator, 2a^-4 * b^5. Divide the coefficients: (1/16) / 2 = 1/32. Divide the variables by subtracting the exponents of like bases: for a, it is -6 - (-4) = -2, and for b, it is 4 - 5 = -1. This yields (1/32) * a^-2 * b^-1. Finally, move the variables with negative exponents to the denominator to attain the fully simplified form.

Step-by-Step Solution

1

Apply the power of a power rule to the expression in the numerator.

(4)^{-2}(a^3)^{-2}(b^{-2})^{-2} = \frac{1}{16}a^{-6}b^4

When an expression in parentheses is raised to a power, the outer exponent must be applied to the numerical coefficient and multiplied by the exponent of each variable inside.

2

Divide the coefficients of the numerator and the denominator.

\frac{1/16}{2} = \frac{1}{16} \times \frac{1}{2} = \frac{1}{32}

The constant factors can be evaluated and divided separately from the variable terms.

3

Apply the quotient of powers rule for the variable bases.

a^{-6 - (-4)} = a^{-2}

and

b^{4 - 5} = b^{-1}

When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

4

Rewrite the expression using only positive exponents.

\frac{1}{32}a^{-2}b^{-1} = \frac{1}{32a^2b}

Final simplified algebraic expressions are standardly written by moving variables with negative exponents to the denominator.

Key Concept

Applying multiple properties of exponents, including the power of a product rule, power of a power rule, and quotient of powers rule with negative integers.

Question 347Question →

A base facility manager purchases a heavy-duty power drill originally priced at $\\$ 85.00 $. He applies a$ \\ $15.50$ discount coupon to the purchase. After the discount is subtracted, a sales tax of $\\$ 0.06$ per dollar is added to the remaining amount. What is the final total cost of the drill?

\\

69.56$

\\

74.60$

\\

73.67$

\\

111.20$

Show answer & explanation

Answer: $\\$ 73.67$

Answer

The final total cost of the drill is

\\

73.67$.

The correct answer accurately subtracts the

\\

15.50

discount from the

\\

85.00

original price to get a subtotal of

\\

69.50

. Then, the tax is correctly calculated by multiplying the subtotal by the decimal rate:

\\

69.50 \times 0.06 = \\

4.17

. Adding the tax to the subtotal yields the final cost of

\\

73.67

.

Step-by-Step Solution

1

Subtract the discount from the original price to find the subtotal.

\\

85.00 - \\

15.50 = \\

69.50$

The sales tax is applied to the remaining amount after the coupon is deducted.

2

Calculate the tax amount by multiplying the subtotal by the tax rate.

\\

69.50 \times 0.06 = \\

4.17

The tax is

\\

0.06$ for every dollar spent in the subtotal.

3

Add the calculated tax to the subtotal to find the final price.

\\

69.50 + \\

4.17 = \\

73.67$

The final total cost must include both the discounted base price and the applied tax.

Key Concept

Applying decimal operations to solve multi-step currency word problems involving discounts and taxes.

Question 348Question →

Sarah works as a wholesale medical supplier. She is compensated with a fixed weekly base pay of $\$750$ . In addition to her base pay, she receives an $8\%$ commission on the portion of her weekly sales that exceeds a $\$5,000$ quota. If her total sales for the week amount to $\$12,500$ , what is her total gross earnings for the week?

\$600

\$810

\$1,350

\$1,750

Show answer & explanation

Answer: $\$1,350$

Answer

\$1,350

To find the total gross earnings, you must first identify the sales amount subject to commission by subtracting the quota from the total sales (

\$12,500 - \$5,000 = \$7,500

). Next, find the

8\%

commission on that amount (

0.08 \times 7,500 = \$600

). Finally, add the commission to the fixed base pay (

\$750 + \$600 = \$1,350

).

Step-by-Step Solution

1

Determine the amount of sales eligible for commission.

\$12,500 - \$5,000 = \$7,500

The commission is only applied to the portion of sales that exceeds the $5,000 quota.

2

Calculate the commission earned.

8\% \text{ of } \$7,500 = 0.08 \times 7,500 = \$600

Multiply the commissionable sales by the commission rate (expressed as a decimal) to find the extra earnings.

3

Calculate the total gross earnings.

\$750 + \$600 = \$1,350

Add the base pay to the earned commission to find the total gross pay for the week.

Key Concept

Calculating gross pay involving a base salary and a commission with a sales quota.

Question 349Question →

During an analysis of a parabolic trajectory, an engineer must factor the polynomial $4x^2 - 4x - 48$ completely to determine its zero points. What is the correctly factored form of this polynomial?

4(x - 4)(x + 3)

4(x + 4)(x - 3)

(4x - 16)(x + 3)

4(x - 6)(x + 2)

Show answer & explanation

Answer: $4(x - 4)(x + 3)$

Answer

The completely factored form of the polynomial is

4(x - 4)(x + 3)

.

To factor completely, first factor out the GCF of 4 to get

4(x^2 - x - 12)

. Then, factor the resulting trinomial by finding two numbers that multiply to -12 and add to -1. Those numbers are -4 and 3, yielding the final form

4(x - 4)(x + 3)

.

Step-by-Step Solution

1

Extract the greatest common factor (GCF) from all terms in the polynomial

4x^2 - 4x - 48

.

4(x^2 - x - 12)

Factoring out the GCF simplifies the polynomial and makes the remaining quadratic expression easier to factor.

2

Find two integers that multiply to the constant term (-12) and add up to the coefficient of the middle term (-1).

The two integers are -4 and +3.

These integers provide the constants needed for the two binomial factors of the quadratic trinomial.

3

Write the final expression by combining the GCF with the factored binomials.

4(x - 4)(x + 3)

This shows the polynomial broken down into its prime, indivisible factors.

Key Concept

Factoring out a Greatest Common Factor (GCF) before factoring a quadratic trinomial.

Question 350Question →

A base recreation center needs to purchase tennis balls for an upcoming tournament. A single can of tennis balls costs $\$4.50$ . A case containing $10$ cans costs $\$40.00$ . How much money is saved per can by purchasing the case instead of individual cans?

\$0.50

\$0.05

\$4.25

\$35.50

Show answer & explanation

Answer: $\$0.50$

Answer

\$0.50

Purchasing the case means each can costs

\$40.00

divided by

10

, which is

\$4.00

per can. Subtracting this bulk unit price from the individual price of

\$4.50

gives a savings of

\$0.50

per can.

Step-by-Step Solution

1

Calculate the unit price of a single can when bought by the case.

\$40.00 \div 10 = \$4.00

per can

To compare prices on equal terms, you must find the cost of one unit for the bulk option.

2

Calculate the difference between the individual can price and the bulk unit price.

\$4.50 - \$4.00 = \$0.50

Subtracting the lower unit price from the higher individual price reveals the exact savings per can.

Key Concept

Comparing the unit rate of a bulk purchase to an individual item price to determine savings.

Estimated Time:45s

Question 351Question →

A refueling truck contains $150$ gallons of aviation fuel. The fuel is distributed among three training aircraft in a ratio of $2:3:5$ . How many quarts of fuel does the second aircraft receive? ( $1\text{ gallon} = 4\text{ quarts}$ )

360

45

180

300

Show answer & explanation

Answer: 180

Answer

180

The total number of ratio parts is

2 + 3 + 5 = 10

. Dividing the total

150

gallons by

10

parts gives

15

gallons per part. The second aircraft receives

3

parts, which is

3 \times 15 = 45

gallons. Converting to quarts,

45 \text{ gallons} \times 4 \text{ quarts/gallon} = 180 \text{ quarts}

.

Step-by-Step Solution

1

Calculate the total number of ratio parts.

2 + 3 + 5 = 10

parts

To find the value of a single part, we must divide the total amount by the sum of the ratio parts.

2

Determine the value of one part in gallons.

150 \div 10 = 15

gallons per part

This establishes the base unit for the ratio distribution.

3

Calculate the second aircraft's share in gallons.

3 \times 15 = 45

gallons

The second aircraft corresponds to the middle term in the

2:3:5

ratio, which is

3

parts.

4

Convert the share from gallons to quarts.

45 \times 4 = 180

quarts

The question specifically asks for the final answer in quarts.

Key Concept

Dividing in a Ratio with Unit Conversions

Question 352Question →

A military cargo plane flies from Base Alpha to Base Bravo at a steady cruising speed of $320\text{ miles per hour}$ . This first leg of the flight takes $2\text{ hours and } 45\text{ minutes}$ . After refueling, the plane continues to Base Charlie, flying at $280\text{ miles per hour}$ for $1\text{ hour and } 30\text{ minutes}$ . What is the total distance traveled by the cargo plane from Base Alpha to Base Charlie? Include both the numeric value and unit in your answer (e.g., '500 miles').

Show answer & explanation

Answer: 1300 miles; 1,300 miles; 1300 mi; 1300; 1,300

Answer

1300 miles

The correct answer is found by first converting all travel times into decimal hours. The first leg is

2.75

hours at

320\text{ mph}

, covering

880\text{ miles}

. The second leg is

1.5

hours at

280\text{ mph}

, covering

420\text{ miles}

. Adding these two distances yields a total of

1300\text{ miles}

.

Step-by-Step Solution

1

Convert the flight times into decimal hours.

2\text{ hours and } 45\text{ minutes} = 2.75\text{ hours}

;

1\text{ hour and } 30\text{ minutes} = 1.5\text{ hours}

.

To calculate distance using miles per hour, the time units must be converted to hours.

2

Calculate the distance for the first leg from Base Alpha to Base Bravo.

320\text{ mph} \times 2.75\text{ hours} = 880\text{ miles}

.

Distance equals rate multiplied by time (

d = r \times t

).

3

Calculate the distance for the second leg from Base Bravo to Base Charlie.

280\text{ mph} \times 1.5\text{ hours} = 420\text{ miles}

.

The second leg must be calculated independently due to the change in both speed and travel time.

4

Add the distances of both legs to find the total distance traveled.

880\text{ miles} + 420\text{ miles} = 1300\text{ miles}

.

The total journey distance is the sum of the distances of all individual legs.

Key Concept

Calculating multi-leg total distance using

d = r \times t

with unit conversions.

Question 353Question →

When the algebraic expression $12y^3 + 12y^2 - 45y$ is factored completely, which of the following expressions is produced?

3y(2y - 3)(2y + 5)

3y(2y + 3)(2y - 5)

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

-3y(2y - 3)(2y + 5)

Show answer & explanation

Answer: $3y(2y - 3)(2y + 5)$

Answer

The completely factored form is

3y(2y - 3)(2y + 5)

.

The correct answer accurately extracts the full GCF of

3y

and correctly factors the remaining trinomial into

(2y - 3)(2y + 5)

. Expanding this result confirms it equals

12y^3 + 12y^2 - 45y

.

Step-by-Step Solution

1

Identify and extract the Greatest Common Factor (GCF) from all terms in

12y^3 + 12y^2 - 45y

.

The GCF is

3y

. Factoring it out leaves

3y(4y^2 + 4y - 15)

.

Extracting the GCF first simplifies the polynomial and is a required step for complete factorization.

2

Factor the remaining non-monic trinomial,

4y^2 + 4y - 15

, by finding two numbers that multiply to

a \times c

and add to

b

.

We need numbers that multiply to

(4)(-15) = -60

and add to

4

. The numbers are

10

and

-6

.

This prepares the trinomial for factoring by grouping.

3

Rewrite the middle term (

4y

) using these two numbers and factor by grouping.

4y^2 + 10y - 6y - 15 = 2y(2y + 5) - 3(2y + 5) = (2y - 3)(2y + 5)

.

Grouping extracts common binomial factors, breaking down the quadratic into its simplest linear binomials.

4

Combine the original GCF with the newly factored binomials.

3y(2y - 3)(2y + 5)

To represent the full and complete factorization of the original polynomial.

Key Concept

Factoring polynomials by first extracting the GCF and subsequently using the ac-method (or grouping) to factor a non-monic quadratic trinomial.

Question 354Question →

Simplify the following polynomial expression completely:

(3x - 2y)(2x^2 - xy + 4y^2) - x(x - 3y)^2

Which of the following represents the fully expanded and simplified result?

5x^3 - x^2y + 5xy^2 - 8y^3

5x^3 - 13x^2y + 23xy^2 - 8y^3

5x^3 - 7x^2y + 23xy^2 - 8y^3

7x^3 - 13x^2y + 23xy^2 - 8y^3

Show answer & explanation

Answer: $5x^3 - x^2y + 5xy^2 - 8y^3$

Answer

The completely simplified expression is

5x^3 - x^2y + 5xy^2 - 8y^3

.

The correct answer is derived by carefully following the order of operations: first expanding both the left

(3x - 2y)(2x^2 - xy + 4y^2)

and right

x(x - 3y)^2

sides of the subtraction independently, and then combining the resulting polynomials. Distributing the negative sign correctly during the final subtraction yields

5x^3 - x^2y + 5xy^2 - 8y^3

.

Step-by-Step Solution

1

Expand the product of the binomial and the trinomial:

(3x - 2y)(2x^2 - xy + 4y^2)

.

6x^3 - 3x^2y + 12xy^2 - 4x^2y + 2xy^2 - 8y^3

Every term in the first binomial must be multiplied by every term in the trinomial to ensure full expansion.

2

Combine the like terms in the first expanded expression.

6x^3 - 7x^2y + 14xy^2 - 8y^3

Combining like terms simplifies the polynomial before performing the final subtraction operation.

3

Expand the squared binomial in the second part of the expression:

(x - 3y)^2

.

x^2 - 6xy + 9y^2

A perfect square binomial

(a-b)^2

expands to

a^2 - 2ab + b^2

.

4

Distribute the

x

into the expanded squared binomial:

x(x^2 - 6xy + 9y^2)

.

x^3 - 6x^2y + 9xy^2

The outer monomial must be distributed to each term in the trinomial.

5

Subtract the second result from the first result:

(6x^3 - 7x^2y + 14xy^2 - 8y^3) - (x^3 - 6x^2y + 9xy^2)

.

6x^3 - 7x^2y + 14xy^2 - 8y^3 - x^3 + 6x^2y - 9xy^2

Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses.

6

Combine all remaining like terms.

5x^3 - x^2y + 5xy^2 - 8y^3

Grouping

x^3

,

x^2y

, and

xy^2

terms produces the final simplified answer in standard form.

Key Concept

Multiplying complex polynomials and applying subtraction across multiple terms

Question 355Question →

A restaurant owner is purchasing bulk cleaning supplies. The regular price for the order is $\$250$ . If the wholesale supplier offers a $10\%$ discount on the entire order, what is the final sale price?

\$247.50

\$225

\$240

\$25

Show answer & explanation

Answer: $\$225$

Answer

The final sale price is

\$225

.

To find the final sale price, first calculate the discount amount by finding

10\%

of

\$250

, which is

\$25

. Then, subtract the discount amount from the original price:

\$250 - \$25 = \$225

.

Step-by-Step Solution

1

Calculate the discount amount by finding

10\%

of the regular price.

0.10 \times 250 = 25

The discount amount must be determined before it can be subtracted from the total.

2

Subtract the discount amount from the regular price.

250 - 25 = 225

The final sale price is the original price minus the discount savings.

Key Concept

Discounts and Sale Prices

Question 356Question →

The polynomial $x^2 + 6x - 16$ can be factored as $(x - 2)(x + k)$ . What is the value of the constant $k$ ?

Show answer & explanation

Answer: 8; +8; eight

Answer

8

The constant

k

must be 8 because in the factored form

(x - 2)(x + k)

, the product of -2 and

k

must equal the constant term of the polynomial, which is -16. Since

-2 \times 8 = -16

and the sum

-2 + 8 = 6

gives the correct middle coefficient, the value of

k

is exactly 8.

Step-by-Step Solution

1

Analyze the relationship between the expanded polynomial and its factored form.

The expanded form is

x^2 + 6x - 16

and the factored form is

(x - 2)(x + k)

.

This sets up the equation to solve for the missing constant

k

.

2

Use the property of multiplying binomials to find the constant term.

The product of the constants in the binomials must equal the constant term of the polynomial:

-2 \times k = -16

.

When expanding

(x + a)(x + b)

, the constant term is always equal to

a \times b

.

3

Solve the equation for

k

.

k = -16 \div (-2) = 8

.

Dividing both sides by -2 isolates

k

.

4

Verify the middle term coefficient with the found value of

k

.

-2 + 8 = 6

, which matches the middle term

6x

.

Checking the sum of the constants ensures the entire factorization is correct.

Key Concept

Factoring quadratic trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient.

Question 357Question →

A surveillance drone flies at a constant speed of $40$ miles per hour. How many minutes will it take the drone to travel a distance of $10$ miles?

15

4

25

0.25

Show answer & explanation

Answer: $15$

Answer

15

minutes

The correct answer is found by dividing the distance (

10

miles) by the speed (

40

miles per hour) to get the time in hours, which is

0.25

hours. Then, converting hours to minutes by multiplying by

60

gives exactly

15

minutes.

Step-by-Step Solution

1

Calculate the time required in hours.

0.25

hours

Time is equal to distance divided by rate. Dividing

10

miles by

40

miles per hour gives

0.25

hours.

2

Convert the time from hours into minutes.

15

minutes

Since there are

60

minutes in an hour, multiplying

0.25

hours by

60

yields

15

minutes.

Key Concept

Calculating elapsed time using the distance formula and converting units of time.

Question 358Question →

When the algebraic expression $5x^2 - 13x - 6$ is factored completely, it can be written in the form $(5x + a)(x - b)$ , where $a$ and $b$ are positive integers. What is the value of $a + b$ ?

Show answer & explanation

Answer: 5; five

Answer

5

Factoring the quadratic

5x^2 - 13x - 6

yields

(5x + 2)(x - 3)

. By comparing this to the target format

(5x + a)(x - b)

, we can identify that

a = 2

and

b = 3

. Adding these together gives the final answer of 5.

Step-by-Step Solution

1

Determine the factoring method for the non-monic quadratic expression

5x^2 - 13x - 6

.

Since the leading coefficient is 5, look for two numbers that multiply to

(5)(-6) = -30

and add to the middle coefficient,

-13

.

This sets up the factor-by-grouping method.

2

Find the two numbers.

The numbers are

-15

and

2

.

(-15) \times 2 = -30

and

-15 + 2 = -13

.

3

Rewrite the middle term using these two numbers and factor by grouping.

5x^2 - 15x + 2x - 6 = 5x(x - 3) + 2(x - 3) = (5x + 2)(x - 3)

.

Grouping common terms extracts the binomial factors.

4

Match the factored form to the given format

(5x + a)(x - b)

.

Comparing

(5x + 2)(x - 3)

to

(5x + a)(x - b)

reveals that

a = 2

and

b = 3

.

Both

a

and

b

correspond exactly to the positive integer components of the binomials.

5

Calculate the final required sum,

a + b

.

2 + 3 = 5

.

The question specifically asks for the sum of the parameters.

Key Concept

Factoring non-monic quadratic trinomials

Question 359Question →

Match each unsimplified algebraic expression with its fully simplified equivalent form. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each unsimplified expression matches exactly one fully simplified form by applying the product, quotient, and power rules of exponents.

The correct matches are found by systematically applying the fundamental properties of exponents: adding exponents for multiplication, subtracting for division, multiplying for powers of powers, and converting negative exponents to reciprocals.

Step-by-Step Solution

1

Simplify the product of powers expression: (3x^-2y^4)(5x^6y^-3)

15x^4y

Multiply numerical coefficients together. Add exponents for variables with the same base according to the product rule.

2

Simplify the quotient of powers expression: (30x^5y^-2) / (2x^9y^3)

15 / (x^4y^5)

Divide numerical coefficients. Subtract the denominator's exponent from the numerator's exponent. Convert negative exponents to positive by moving them to the denominator.

3

Simplify the power of a product expression: (-2x^3y^-1)^4

(16x^12) / y^4

Raise every factor inside the parentheses to the 4th power. Note that an even power of a negative coefficient yields a positive result.

4

Simplify the expression with an outer negative exponent: (3x^-2 / y^4)^-2

(x^4y^8) / 9

Multiply all inner exponents by the outer exponent -2. Then take the reciprocal of terms with negative exponents to rewrite them with positive exponents.

Key Concept

Operations with Variable Exponents

Question 360Question →

A combat medic is restocking his medical kit. He purchases $5$ rolls of medical tape for $\$2.45$ each and $3$ bottles of antiseptic for $\$4.20$ each. If he pays with a $\$50.00$ bill and there is no sales tax, what is the exact amount of change he receives?

Show answer & explanation

Answer: 25.15

Answer

25.15

The correct answer is found by calculating the total cost of all items (

\$12.25 + \$12.60 = \$24.85

) and then subtracting that total from the

\$50.00

payment amount to find the change due (

\$50.00 - \$24.85 = \$25.15

).

Step-by-Step Solution

1

Calculate the total cost of the medical tape.

$12.25

Multiply the quantity (5) by the unit price ($2.45).

2

Calculate the total cost of the antiseptic.

$12.60

Multiply the quantity (3) by the unit price ($4.20).

3

Add the costs of the tape and antiseptic to find the total purchase amount.

$24.85

12.25 +

12.60 = $24.85.

4

Subtract the total purchase amount from the payment amount.

$25.15

50.00 -

24.85 = $25.15. This is the exact change received.

Key Concept

Decimal Operations and Currency Word Problems

Answer

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