All practice questions

Question 41Question →

At Oakridge High School, $\frac{3}{5}$ of the students participate in extracurricular sports. Of the students who participate in extracurricular sports, $25\%$ are members of the track and field team. What percent of the total students at Oakridge High School are on the track and field team?

Show answer & explanation

Answer: 15

Answer

The correct answer is 15.

To find the portion of the total students on the track team, convert the fraction of students in sports to a decimal, which is 0.60. Then, multiply this by the track team percentage expressed as a decimal, which is 0.25. The product is 0.15, which represents 15% of the total student body.

Step-by-Step Solution

1

Convert the fraction of students playing sports to a decimal

0.60

To easily multiply the two portions, convert the fraction to a decimal:

\frac{3}{5} = 0.60

.

2

Multiply the sports-participating portion by the track team percentage

0.15

To find a percentage of a decimal, convert the percentage to a decimal (

25\% = 0.25

) and multiply:

0.60 \times 0.25 = 0.15

.

3

Convert the final decimal back to a percentage

15\%

Multiply the decimal by 100 to express the final portion as a percentage:

0.15 \times 100 = 15\%

.

Key Concept

Fractions, Decimals, and Percentages

Estimated Time:45s

Question 42Question →

The prime factorization of a positive integer $N$ is of the form $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are non-negative integers. The number $N$ has exactly $12$ positive factors. If $N$ is a multiple of $4$ but is not divisible by $3$ , what is the smallest possible value of $N$ ?

Show answer & explanation

Answer: 160

Answer

The smallest possible value of

N

is 160.

The correct value is 160 because it satisfies all the conditions: its prime factorization has only 2 and 5 (no 3, so not divisible by 3), the exponent of 2 is 5 (which is greater than or equal to 2, so it is a multiple of 4), the number of factors is

(5+1)(1+1) = 12

, and it is the smallest such integer.

Step-by-Step Solution

1

Set up the factor counting formula using the prime factorization.

The number of positive factors of

N = 2^a \times 3^b \times 5^c

is

(a+1)(b+1)(c+1) = 12

.

The number of positive factors of any integer is found by adding one to each exponent in its prime factorization and multiplying the results.

2

Apply the given divisibility conditions to simplify the exponents.

Since

N

is not divisible by

3

,

b = 0

. Since

N

is a multiple of

4

,

a \ge 2

. The equation simplifies to

(a+1)(c+1) = 12

with

a+1 \ge 3

.

A number is not divisible by a prime if its exponent in the prime factorization is 0. A number is a multiple of 4 if the exponent of 2 in its prime factorization is at least 2.

3

List all possible pairs of

(a+1, c+1)

that multiply to 12 where

a+1 \ge 3

, and compute the corresponding values of

N

.

Case 1:

a+1 = 3 \implies a = 2, c = 3 \implies N = 2^2 \times 5^3 = 500

. Case 2:

a+1 = 4 \implies a = 3, c = 2 \implies N = 2^3 \times 5^2 = 200

. Case 3:

a+1 = 6 \implies a = 5, c = 1 \implies N = 2^5 \times 5^1 = 160

. Case 4:

a+1 = 12 \implies a = 11, c = 0 \implies N = 2^{11} \times 5^0 = 2048

.

Testing all possible divisor pairs of 12 that satisfy the constraint on

a

allows us to find all possible candidate values for

N

.

4

Determine the smallest value of

N

from the candidate values.

The smallest candidate is 160.

Comparing the values 500, 200, 160, and 2048 shows that 160 is the minimum.

Key Concept

Factors, Multiples, and Prime Factorization

Question 43Question →

Three distinct integers, $a$ , $b$ , and $c$ , lie on a standard number line such that $a < b < c$ . The distance between $a$ and $b$ is $3$ times the distance between $b$ and $c$ . If $|a| = 15$ , $|c| = 7$ , and $b < 0$ , what is the value of $b$ ?

Show answer & explanation

Answer: -9

Answer

The value of

b

is

-9

.

The correct answer is

-9

. By interpreting

|a| = 15

and

|c| = 7

with the constraint

a < b < c

, we find

a = -15

. Testing the possible values for

c

, when

c = -7

, we set up the distance equation

b - (-15) = 3(-7 - b)

, which simplifies to

b + 15 = -21 - 3b

, giving

4b = -36

and

b = -9

. This satisfies all constraints, including

b

being a negative integer.

Step-by-Step Solution

1

Find the possible coordinates for

a

and

c

based on their absolute values.

a \in \{-15, 15\}

and

c \in \{-7, 7\}

.

The absolute value of a number represents its distance from zero, so

|x| = d \implies x = \pm d

.

2

Use the ordering condition

a < b < c

to eliminate invalid combinations.

a = -15

and

c \in \{-7, 7\}

.

Since

a

must be less than

c

,

a

cannot be

15

because both possible values of

c

(

-7

and

7

) are less than

15

.

3

Set up an equation representing the distance relationship on the number line.

b + 15 = 3(c - b) \implies 4b = 3c - 15

.

For points on a number line ordered

a < b < c

, the distance between

a

and

b

is

b - a

, and the distance between

b

and

c

is

c - b

.

4

Substitute each possible value of

c

and solve for

b

to find the one that yields a negative integer.

For

c = -7

,

b = -9

.

If

c = 7

,

b = 1.5

, which is not an integer. If

c = -7

,

b = -9

, which is a negative integer, satisfying all given conditions.

Key Concept

Using absolute values and relative order to determine integer positions and distances on a number line.

Question 44Question →

Three lighthouse beacons flash at regular intervals. Beacon A flashes every $24$ seconds, Beacon B flashes every $36$ seconds, and Beacon C flashes every $s$ seconds, where $s$ is a positive integer. If all three beacons flash at the same instant, and the next time they all flash at the same instant is exactly $6$ minutes later, what is the number of possible values for $s$ ?

Show answer & explanation

Answer: 12

Answer

12

To find the number of possible values for

s

, we convert the joint flashing time to seconds (

6 \text{ minutes} = 360 \text{ seconds}

). The least common multiple (LCM) of the three intervals must equal this joint interval:

\text{LCM}(24, 36, s) = 360

. Writing the prime factorizations gives

24 = 2^3 \cdot 3^1

,

36 = 2^2 \cdot 3^2

, and

360 = 2^3 \cdot 3^2 \cdot 5^1

. For any positive integer

s = 2^a \cdot 3^b \cdot 5^c

, the power of each prime in the LCM is the maximum of the powers in the individual prime factorizations. This gives the constraints:

a \in \{0, 1, 2, 3\}

(4 options),

b \in \{0, 1, 2\}

(3 options), and

c = 1

(1 option). Multiplying these possibilities gives

4 \times 3 \times 1 = 12

possible values for

s

.

Step-by-Step Solution

1

Convert the joint interval to seconds.

360

seconds

The individual intervals are given in seconds, so the joint interval must be converted to the same unit to perform calculations.

2

Set up the LCM equation.

\text{LCM}(24, 36, s) = 360

The beacons will flash together at intervals that are multiples of all three individual intervals. The first time they flash together again represents the least common multiple.

3

Find the prime factorizations of the known numbers.

24 = 2^3 \cdot 3^1

,

36 = 2^2 \cdot 3^2

, and

360 = 2^3 \cdot 3^2 \cdot 5^1

Prime factorization allows us to analyze the relationship between the individual numbers and their least common multiple.

4

Analyze the exponents of the prime factors of

s

.

For

s = 2^a \cdot 3^b \cdot 5^c

, we must have

a \in \{0, 1, 2, 3\}

,

b \in \{0, 1, 2\}

, and

c = 1

.

The exponent of each prime factor in the LCM is the maximum of the exponents of that prime factor in the numbers being combined. Since the LCM has

2^3

, the maximum exponent of 2 must be 3, which is already satisfied by

24 = 2^3 \cdot 3^1

, so

a

can be any integer from 0 to 3. Since the LCM has

3^2

, the maximum exponent of 3 must be 2, which is already satisfied by

36 = 2^2 \cdot 3^2

, so

b

can be any integer from 0 to 2. Since the LCM has

5^1

and neither 24 nor 36 has a factor of 5,

s

must provide exactly one factor of 5 (

c=1

).

5

Calculate the total number of combinations for

s

.

4 \times 3 \times 1 = 12

Since the choice of each exponent is independent, we multiply the number of choices for each prime factor's exponent.

Key Concept

Using prime factorizations to determine the relationship between numbers and their least common multiple (LCM).

Question 45Question →

A store sells a winter coat that is originally priced at $120. During a weekend clearance sale, the coat's price is reduced by 25%. What is the sale price of the coat, in dollars?

Show answer & explanation

Answer: 90

Answer

The sale price of the coat is $90.

The correct answer is

90. To find the sale price, we compute the discount by taking 25% of

120, which is

30, and then subtract that discount from the original price of

120.

Step-by-Step Solution

1

Find 25% of $120 to determine the discount amount.

$30

A 25% reduction means we calculate

120 multiplied by 0.25, which equals

30.

2

Subtract the discount from the original price.

$90

The sale price is the original price minus the discount:

120 -

30 = $90.

Key Concept

Calculating a percentage markdown to determine a final price

Estimated Time:45s

Question 46Question →

A positive integer $N$ has a prime factorization of the form $p^2 \times q$ , where $p$ and $q$ are distinct prime numbers. If the sum of all the positive factors of $N$ (including $1$ and $N$ ) is $78$ , what is the value of $N$ ?

Show answer & explanation

Answer: 45

Answer

The value of

N

is 45.

By applying the sum of divisors formula, the sum of factors of

N = p^2 \times q

is represented as

(1 + p + p^2)(1 + q) = 78

. Factoring 78 into two integers that satisfy the prime constraints of

p \ge 2

and

q \ge 2

yields

13 \times 6 = 78

. Solving

1 + p + p^2 = 13

gives

p = 3

, and

1 + q = 6

gives

q = 5

. Since 3 and 5 are distinct primes,

N = 3^2 \times 5 = 45

.

Step-by-Step Solution

1

Express the sum of factors of

N

algebraically

(1 + p + p^2)(1 + q) = 78

The sum of all positive factors of a number with prime factorization

p^a q^b

is given by the product of the sums of the powers of each prime factor.

2

Determine the constraints on

p

and

q

based on them being prime numbers

1 + p + p^2 \ge 7

and

1 + q \ge 3

The smallest prime number is 2, so

p \ge 2

and

q \ge 2

.

3

Find the factor pairs of 78 that satisfy the constraints

(1 + p + p^2, 1 + q) \in \{(13, 6), (26, 3)\}

The factors of 78 are 1, 2, 3, 6, 13, 26, 39, 78. We pair them such that one factor is at least 7 and the other is at least 3.

4

Solve for

p

and

q

for each possible factor pair

p = 3

and

q = 5

If

1 + p + p^2 = 13

, then

p^2 + p - 12 = 0

, which solves to

p = 3

(since

p > 0

). This leaves

1 + q = 6 \implies q = 5

. Both 3 and 5 are distinct primes. The other case,

1 + p + p^2 = 26

, has no integer solution for

p

.

5

Calculate the value of

N

N = 3^2 \times 5 = 45

Substitute the prime values back into the expression for

N

.

Key Concept

Sum of Divisors Formula and Prime Factorization

Estimated Time:1m 30s

Question 47Question →

On a standard number line, the coordinate of point $P$ is an integer $x$ . If the sum of the distances from $P$ to $-2$ and from $P$ to $4$ is equal to the distance from $P$ to $10$ , what is the sum of all possible values of $x$ ?

Show answer & explanation

Answer: -4

Answer

The sum of all possible values of the integer

x

is

-4

.

The correct answer is

-4

. The distance between any two coordinates

a

and

b

on a number line is defined as

|a - b|

. Translating the problem gives the equation

|x + 2| + |x - 4| = |x - 10|

. Breaking the number line into intervals around the critical points

x = -2

,

x = 4

, and

x = 10

yields two valid integer solutions:

x = -8

and

x = 4

. The sum of these values is

-8 + 4 = -4

.

Step-by-Step Solution

1

Express the distances on the number line using absolute values.

The distance from

P(x)

to

-2

is

|x - (-2)| = |x + 2|

. The distance from

P(x)

to

4

is

|x - 4|

. The distance from

P(x)

to

10

is

|x - 10|

. The equation is

|x + 2| + |x - 4| = |x - 10|

.

The absolute value

|a - b|

represents the distance between points

a

and

b

on a standard number line.

2

Solve the equation by testing the intervals defined by the critical points

x = -2

,

x = 4

, and

x = 10

.

We analyze the four intervals:
- For

x < -2

:

-(x + 2) - (x - 4) = -(x - 10) \implies -2x + 2 = -x + 10 \implies x = -8

. Since

-8 < -2

, this is a valid solution.
- For

-2 \leq x < 4

:

(x + 2) - (x - 4) = -(x - 10) \implies 6 = -x + 10 \implies x = 4

. Since

4

is not in

[-2, 4)

, there is no solution in this interval.
- For

4 \leq x < 10

:

(x + 2) + (x - 4) = -(x - 10) \implies 2x - 2 = -x + 10 \implies 3x = 12 \implies x = 4

. Since

4

is in

[4, 10)

, this is a valid solution.
- For

x \geq 10

:

(x + 2) + (x - 4) = x - 10 \implies 2x - 2 = x - 10 \implies x = -8

. Since

-8 < 10

, there is no solution in this interval.

Absolute value terms change sign at their critical points, requiring case-by-case evaluation.

3

Sum all valid integer solutions.

The valid values for

x

are

-8

and

4

. Their sum is

-8 + 4 = -4

.

The problem asks for the sum of all possible values of

x

.

Key Concept

Representing geometric distances on a number line using absolute value equations and solving them using interval analysis.

Question 48Question →

A red blood cell has a diameter of approximately $0.000008$ meters. When this number is written in scientific notation as $a \times 10^n$ , where $1 \leq a < 10$ , what is the value of $n$ ?

Show answer & explanation

Answer: -6

Answer

The value of the exponent is

-6

.

To write the number

0.000008

in scientific notation, we shift the decimal point 6 places to the right to get

8

. Since the original value is less than 1, the exponent is negative, giving

8 \times 10^{-6}

. The value of

n

is therefore

-6

.

Step-by-Step Solution

1

Locate the decimal point and determine how many places it must be shifted to obtain a coefficient between 1 and 10.

The decimal point must be shifted 6 places to the right to get the number

8

.

Scientific notation requires the lead coefficient

a

to satisfy

1 \leq a < 10

.

2

Determine the sign and value of the exponent based on the decimal shift.

The exponent

n

is

-6

.

Moving the decimal point to the right to write a decimal value less than 1 results in a negative exponent equal to the number of shifts.

Key Concept

Scientific Notation for Decimals Less Than One

Question 49Question →

At the start of the fiscal year, a city's municipal budget is divided among three departments: Education, Healthcare, and Infrastructure. Education receives $\frac{3}{8}$ of the total budget, Healthcare receives $40\%$ of the total budget, and Infrastructure receives the remainder. Midyear, the total municipal budget is increased by $25\%$ . The budget allocated to Education is increased by $20\%$ of its original amount, and the budget allocated to Healthcare is increased by $15\%$ of its original amount. What percent of the new total municipal budget is allocated to Infrastructure?

Show answer & explanation

Answer: 27.2

Answer

27.2

The correct answer is

27.2\%

. By representing the initial total budget as

1

, we find the initial Infrastructure allocation is

1 - (0.375 + 0.40) = 0.225

. After the budget increase of

25\%

, the new total budget is

1.25

. The new Education allocation is

0.375 \times 1.20 = 0.45

, and the new Healthcare allocation is

0.40 \times 1.15 = 0.46

. The remaining amount for Infrastructure is

1.25 - 0.45 - 0.46 = 0.34

. Expressing

0.34

as a percentage of the new total budget

1.25

gives

\frac{0.34}{1.25} \times 100\% = 27.2\%

.

Step-by-Step Solution

1

Convert the initial budget allocations to decimal shares of the original total budget.

Education share is

0.375

, Healthcare share is

0.40

, and Infrastructure share is

0.225

.

Expressing all initial shares as decimals relative to a total budget of

1.0

makes successive calculations straightforward.

2

Calculate the updated budget amounts for the entire city, Education, and Healthcare.

The new total budget is

1.25

, the new Education allocation is

0.375 \times 1.20 = 0.45

, and the new Healthcare allocation is

0.40 \times 1.15 = 0.46

.

Adjust each allocation by its respective percentage change to find its share relative to the initial total.

3

Compute the remaining budget share left for Infrastructure.

New Infrastructure share is

1.25 - (0.45 + 0.46) = 0.34

of the initial budget.

The sum of the three department budgets must equal the new total budget of

1.25

.

4

Find the percentage of the new total budget represented by the new Infrastructure allocation.

\frac{0.34}{1.25} \times 100\% = 27.2\%

Divide the new Infrastructure share by the new total budget to find the new ratio, then convert to a percentage.

Key Concept

Converting between fractions, decimals, and percentages, and tracking changes across multiple base values.

Alternative Method

Instead of using

1.0

as the base, you can assume an initial total budget of

800

dollars (since

800

is a multiple of

8

, which simplifies

\frac{3}{8}

). Initial Education =

300

, Healthcare =

320

, Infrastructure =

180

. The new total budget is

800 \times 1.25 = 1000

. New Education =

300 \times 1.20 = 360

. New Healthcare =

320 \times 1.15 = 368

. New Infrastructure =

1000 - (360 + 368) = 272

. Percentage =

\frac{272}{1000} \times 100\% = 27.2\%

.

Estimated Time:3m 0s

Question 50Question →

If $a = 2 \times 10^3$ and $b = 3 \times 10^{-4}$ , what is the value of the expression $\sqrt{\frac{a^3 \cdot b^2}{1.8}}$ ?

Show answer & explanation

Answer: 20

Answer

20

Evaluating the components of the expression sequentially:

a^3 = 8 \times 10^9

and

b^2 = 9 \times 10^{-8}

. Multiplying them yields

(8 \times 9) \times 10^{9-8} = 72 \times 10^1 = 720

. Dividing by the decimal

1.8

results in

720 / 1.8 = 400

. The square root of

400

yields

20

.

Step-by-Step Solution

1

Calculate the value of

a^3

8 \times 10^9

Apply the power of a product rule,

(xy)^n = x^n y^n

, and the power of a power rule,

(10^p)^q = 10^{p \cdot q}

.

2

Calculate the value of

b^2

9 \times 10^{-8}

Apply the power of a product rule and power of a power rule to

(3 \times 10^{-4})^2

.

3

Multiply

a^3

and

b^2

720

Multiply the coefficients (

8 \times 9 = 72

) and add the exponents of the base 10 (

10^9 \times 10^{-8} = 10^1

), resulting in

72 \times 10 = 720

.

4

Divide the product by

1.8

400

Substitute the values to get

720 / 1.8

, which simplifies to

7200 / 18 = 400

.

5

Take the square root of the quotient

20

Find the non-negative square root of

400

, which is

20

.

Key Concept

Simplifying numerical expressions containing combinations of exponent rules, roots, and scientific notation.

Estimated Time:1m 30s

Question 51Question →

A scientist estimates that the population of a certain bacteria culture doubles every 4 hours. If the culture starts with $3.0 \times 10^5$ bacteria, the population after 24 hours can be written in scientific notation as $a \times 10^7$ . What is the value of $a$ ?

Show answer & explanation

Answer: 1.92

Answer

The value of the coefficient

a

is

1.92

.

The correct answer is

1.92

. In 24 hours, a population that doubles every 4 hours undergoes 6 doubling cycles. The growth factor is

2^6 = 64

. Multiplying the initial population of

3.0 \times 10^5

by 64 yields

192 \times 10^5

. Converting this value into standard scientific notation gives

1.92 \times 10^7

. Comparing this expression to the format

a \times 10^7

shows that the coefficient

a

equals

1.92

.

Step-by-Step Solution

1

Determine the number of doubling periods.

6 doubling periods

Since the population doubles every 4 hours, over a span of 24 hours it will double

24 \div 4 = 6

times.

2

Calculate the growth multiplier.

64

Doubling 6 times is represented by the exponential expression

2^6

, which evaluates to 64.

3

Multiply the initial population by the growth factor.

192 \times 10^5

Multiply the coefficient of the initial population by the growth multiplier:

3.0 \times 64 = 192

, yielding a total population of

192 \times 10^5

.

4

Convert the resulting value to the required scientific notation format.

1.92 \times 10^7

To express

192 \times 10^5

in the standard scientific notation format

a \times 10^7

, divide 192 by 100 to get 1.92, and multiply the power of 10 by

10^2

(raising the exponent from 5 to 7).

Key Concept

Evaluating exponential growth and converting numbers to standard scientific notation format

Estimated Time:1m 30s

Question 52Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = (3 - 2i)^2 - 4i^{103}$ . What is the imaginary part of $z$ ?

Show answer & explanation

Answer: -8

Answer

The imaginary part of

z

is

-8

.

The expression

(3-2i)^2

expands to

9 - 12i + 4i^2 = 5 - 12i

. The term

i^{103}

simplifies to

-i

since

103

leaves a remainder of

3

when divided by

4

. Subtracting

4i^{103}

corresponds to adding

4i

, giving

z = (5-12i) + 4i = 5-8i

. The coefficient of the imaginary part is

-8

.

Step-by-Step Solution

1

Expand

(3 - 2i)^2

5 - 12i

Use the binomial expansion formula and substitute

i^2 = -1

.

2

Simplify the term

-4i^{103}

4i

Since

103

divided by

4

leaves a remainder of

3

,

i^{103} = i^3 = -i

. Therefore,

-4i^{103} = -4(-i) = 4i

.

3

Combine terms to find

z

and identify its imaginary part

-8

Add the components:

z = (5 - 12i) + 4i = 5 - 8i

. The imaginary part is the coefficient of

i

, which is

-8

.

Key Concept

Operations on complex numbers including binomial expansion, powers of the imaginary unit

i

, and identification of the imaginary part.

Question 53Question →

For the imaginary unit $i = \sqrt{-1}$ , the complex number $w$ is defined as $w = \frac{5 + 12i}{(1 - i)^4}$ . What is the absolute value of $w$ ?

Show answer & explanation

Answer: 3.25

Answer

The absolute value of

w

is 3.25.

The correct answer is 3.25 because simplifying the denominator yields

(1-i)^4 = -4

. Dividing the numerator by

-4

gives the complex number

w = -1.25 - 3i

. The absolute value of

w

is then calculated as

\sqrt{(-1.25)^2 + (-3)^2} = \sqrt{1.5625 + 9} = \sqrt{10.5625} = 3.25

. Alternatively, using properties of absolute values, the absolute value of the quotient is the quotient of the absolute values:

|w| = \frac{|5 + 12i|}{|1-i|^4} = \frac{\sqrt{5^2 + 12^2}}{(\sqrt{1^2 + (-1)^2})^4} = \frac{13}{4} = 3.25

.

Step-by-Step Solution

1

Simplify the denominator

(1 - i)^4

(1 - i)^4 = -4

Calculate

(1 - i)^2 = -2i

, then square the result to obtain

(-2i)^2 = -4

.

2

Write the complex number

w

in standard form

a + bi

w = -1.25 - 3i

Divide each term in the numerator by the simplified denominator

-4

.

3

Calculate the magnitude

|w|

|w| = 3.25

Use the definition of absolute value of a complex number,

|a + bi| = \sqrt{a^2 + b^2}

.

Key Concept

Absolute value of a complex number and operations on complex numbers

Estimated Time:2m 0s

Question 54Question →

If $(5 - 2i) - (2 + ki) = 3 + 4i$ , where $i = \sqrt{-1}$ and $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: -6

Answer

The value of the constant

k

is

-6

.

Distributing the subtraction sign across the second complex number gives

(5 - 2i) - 2 - ki = 3 + 4i

. Combining the real parts (

5 - 2 = 3

) and grouping the imaginary parts yields

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

. Since the real parts are equal, we set the coefficients of the imaginary parts equal to each other:

-2 - k = 4

. Solving for

k

gives

k = -6

.

Step-by-Step Solution

1

Distribute the negative sign to the expression

(2 + ki)

-2 - ki

To remove the parentheses, the subtraction must apply to all terms inside the parentheses.

2

Group and combine the real parts and imaginary parts on the left side of the equation

3 - (2 + k)i

Grouping like terms allows us to express the left side as a standard complex number

a + bi

.

3

Equate the imaginary parts from both sides of the equation

-2 - k = 4

For two complex numbers to be equal, their corresponding real parts and imaginary parts must be equal.

4

Solve the linear equation for

k

k = -6

Isolating

k

gives the value that satisfies the original equation.

Key Concept

Equality of complex numbers and operations of addition/subtraction on complex numbers.

Estimated Time:45s

Question 55Question →

The polynomial $x^3 + 5x^2 - 9x - 45$ can be factored completely into three linear factors of the form $(x - a)(x - b)(x - c)$ , where $a$ , $b$ , and $c$ are integers such that $a < b < c$ . What is the value of $a - b + c$ ?

Show answer & explanation

Answer: 1

Answer

The correct answer is 1.

Factoring the polynomial by grouping gives

(x^2 - 9)(x + 5)

, which simplifies to

(x - 3)(x + 3)(x + 5)

after factoring the difference of squares. Writing this expression in the form

(x - a)(x - b)(x - c)

identifies the values as

3

,

-3

, and

-5

. Ordering these values to satisfy the inequality

a < b < c

yields

a = -5

,

b = -3

, and

c = 3

. Evaluating

a - b + c

gives

-5 - (-3) + 3 = 1

.

Step-by-Step Solution

1

Group the terms of the polynomial

x^3 + 5x^2 - 9x - 45

.

(x^3 + 5x^2) - (9x + 45)

Grouping allows factoring by finding common binomial terms in a cubic polynomial.

2

Factor out the greatest common factor (GCF) from each grouped term.

x^2(x + 5) - 9(x + 5)

The GCF of the first group is

x^2

and the GCF of the second group is

9

.

3

Factor out the common binomial factor

(x + 5)

.

(x^2 - 9)(x + 5)

Both terms share the common factor

(x + 5)

.

4

Factor the quadratic term

x^2 - 9

as a difference of squares.

(x - 3)(x + 3)(x + 5)

x^2 - 9

is a difference of squares, which factors into

(x - 3)(x + 3)

.

5

Rewrite the factors in the form

(x - a)(x - b)(x - c)

to identify the values of the constants.

(x - 3)(x - (-3))(x - (-5))

which gives the set of values

\{3, -3, -5\}

.

Matching the signs of the given form

(x - \text{constant})

is necessary to correctly identify the values of the constants.

6

Sort the values in ascending order to satisfy

a < b < c

.

a = -5

,

b = -3

, and

c = 3

The inequality constraint requires sorting the values from smallest to largest.

7

Calculate the value of the expression

a - b + c

.

-5 - (-3) + 3 = 1

Substitute the sorted values into the target expression.

Key Concept

Factoring a cubic polynomial by grouping and difference of squares, and identifying algebraic constants under inequality constraints.

Question 56Question →

When the polynomial $6x^2 - 11x - 10$ is factored into the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a$ and $c$ are positive and $a > c$ , what is the value of $ad - bc$ ?

Show answer & explanation

Answer: -19

Answer

The value of

ad - bc

is

-19

.

The correct answer is

-19

. Factoring the polynomial

6x^2 - 11x - 10

yields

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

. Under the constraints that

a

and

c

are positive and

a > c

, we must have

a = 3

,

b = 2

,

c = 2

, and

d = -5

. Evaluating the expression

ad - bc

gives

(3)(-5) - (2)(2) = -15 - 4 = -19

.

Step-by-Step Solution

1

Factor the quadratic expression

6x^2 - 11x - 10

.

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

Factoring by grouping is used to rewrite the quadratic trinomial.

2

Apply the positive coefficient constraints and

a > c

to identify the constants.

a = 3

,

b = 2

,

c = 2

,

d = -5

Since the leading coefficients must be positive and

a > c

, we assign

a = 3

from the first factor and

c = 2

from the second factor.

3

Evaluate the expression

ad - bc

.

-19

Substitute the values of

a

,

b

,

c

, and

d

to calculate the final numerical value.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

where

a > 1

Question 57Question →

When the polynomial $6x^3 - 7x^2 - 16x + 12$ is factored completely into three linear factors of the form $(ax + b)(cx + d)(ex + f)$ , where $a$ , $c$ , and $e$ are positive integers, what is the value of $a + b + c + d + e + f$ ?

Show answer & explanation

Answer: 5

Answer

The value of the sum of the coefficients is 5.

The polynomial factors completely over the integers as

(x - 2)(2x + 3)(3x - 2)

. The sum of the six coefficients is

1 + (-2) + 2 + 3 + 3 + (-2) = 5

.

Step-by-Step Solution

1

Find one linear factor of the cubic polynomial using the Factor Theorem.

The root

x = 2

satisfies the equation, so

(x - 2)

is a factor.

Testing integer factors of the constant term 12 reveals that

x = 2

evaluates the polynomial to 0.

2

Perform synthetic division or polynomial long division to divide the cubic by the linear factor.

The quotient is the quadratic expression

6x^2 + 5x - 6

.

This reduces the degree of the polynomial to allow quadratic factoring techniques.

3

Factor the quadratic quotient into two linear binomials.

The quadratic factors into

(2x + 3)(3x - 2)

.

Using the AC method,

6 \times (-6) = -36

, and the factors of

-36

that sum to

5

are

9

and

-4

.

4

Identify the coefficients and sum them.

The sum is

1 + (-2) + 2 + 3 + 3 + (-2) = 5

.

The factors are

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

, corresponding to the coefficients

a=1, b=-2, c=2, d=3, e=3, f=-2

.

Key Concept

Complete factorization of cubic polynomials with integer coefficients using the Rational Root Theorem and quadratic factoring.

Question 58Question →

Solve the equation $\sqrt{2x + 7} - x = -4$ for $x$ . What is the value of the real solution?

Show answer & explanation

Answer: 9

Answer

The only real solution to the equation is 9.

Isolating the radical yields

\sqrt{2x + 7} = x - 4

. Squaring both sides results in the quadratic equation

2x + 7 = x^2 - 8x + 16

, which simplifies to

x^2 - 10x + 9 = 0

. Factoring gives

(x - 9)(x - 1) = 0

, yielding potential solutions of 9 and 1. Checking these solutions in the original equation reveals that 9 is valid, while 1 is extraneous. Therefore, the correct real solution is 9.

Step-by-Step Solution

1

Isolate the radical on one side of the equation.

\sqrt{2x + 7} = x - 4

Before squaring both sides, the radical term must be isolated to simplify the algebraic manipulation.

2

Square both sides of the equation.

2x + 7 = (x - 4)^2

Squaring both sides eliminates the square root.

3

Expand the squared binomial.

2x + 7 = x^2 - 8x + 16

Applying the identity

(a - b)^2 = a^2 - 2ab + b^2

is necessary to write the equation in polynomial form.

4

Set the quadratic equation to zero.

x^2 - 10x + 9 = 0

Subtracting

2x

and

7

from both sides allows us to solve the quadratic equation.

5

Factor the quadratic equation.

(x - 9)(x - 1) = 0

, giving potential solutions

x = 9

or

x = 1

.

Finding the roots of the quadratic equation provides the candidate solutions.

6

Verify candidates in the original equation.

The solution

x = 9

is valid, while

x = 1

is extraneous.

Squaring both sides can introduce extraneous solutions, so each candidate must be checked in the original equation.

Key Concept

Solving radical equations and verifying for extraneous solutions

Question 59Question →

If the polynomial $12x^2 + 10x - 8$ is factored completely into the form $k(ax - 1)(bx + c)$ , where $k$ , $a$ , $b$ , and $c$ are positive integers, what is the value of $k + a + b + c$ ?

Show answer & explanation

Answer: 11

Answer

The value of

k + a + b + c

is

11

.

To factor the polynomial

12x^2 + 10x - 8

completely, we first factor out the greatest common factor of

2

, yielding

2(6x^2 + 5x - 4)

. Next, we factor the quadratic trinomial

6x^2 + 5x - 4

by finding two numbers that multiply to

6 \times (-4) = -24

and add to

5

. These numbers are

8

and

-3

. Splitting the linear term and factoring by grouping gives

6x^2 + 8x - 3x - 4 = 2x(3x + 4) - 1(3x + 4) = (2x - 1)(3x + 4)

. The completely factored expression is

2(2x - 1)(3x + 4)

. Comparing this with

k(ax - 1)(bx + c)

where

k, a, b, c

are positive integers, we determine that

k = 2

,

a = 2

,

b = 3

, and

c = 4

. Summing these values gives

2 + 2 + 3 + 4 = 11

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from the terms of the polynomial.

2(6x^2 + 5x - 4)

Factoring out the greatest common factor simplifies the coefficients, making the quadratic trinomial easier to factor.

2

Find two integers that multiply to

ac = 6 \times (-4) = -24

and add to

b = 5

.

The two numbers are

8

and

-3

.

These integers are needed to split the linear term in order to factor the quadratic by grouping.

3

Rewrite the middle term and factor the trinomial by grouping.

(2x - 1)(3x + 4)

Rewriting the trinomial as

6x^2 + 8x - 3x - 4

allows grouping of the first two terms

2x(3x + 4)

and the last two terms

-1(3x + 4)

to extract the common binomial factor.

4

Combine the factors and match the coefficients to the form

k(ax - 1)(bx + c)

.

k = 2

,

a = 2

,

b = 3

, and

c = 4

The completely factored expression is

2(2x - 1)(3x + 4)

. Matching this to the given template where all constants are positive integers yields

k = 2

,

a = 2

,

b = 3

, and

c = 4

.

5

Calculate the sum of the constants.

11

Adding the values gives

2 + 2 + 3 + 4 = 11

.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

after removing a greatest common factor.

Question 60Question →

A rectangular garden is surrounded by a uniform gravel path that is $1$ foot wide. The length of the garden is $3$ feet less than twice its width. If the total area of the garden and the path combined is $117$ square feet, what is the width of the garden, in feet?

Show answer & explanation

Answer: 7

Answer

The width of the garden is 7 feet.

By representing the garden's width as

w

, the length is

2w - 3

. The combined dimensions including the 1-foot uniform path on all sides are

w + 2

and

2w - 1

. Setting their product equal to the combined area of 117 square feet gives

(w+2)(2w-1) = 117

, which simplifies to the quadratic equation

2w^2 + 3w - 119 = 0

. Factoring this equation yields

(2w + 17)(w - 7) = 0

, giving the solutions

w = -8.5

and

w = 7

. Since width must be positive, the width of the garden is

7

feet.

Step-by-Step Solution

1

Define variables for the garden's dimensions and the combined dimensions including the path.

Garden width =

w

, garden length =

2w - 3

. Combined width =

w + 2

, combined length =

2w - 1

.

The path surrounds the garden uniformly, adding

1

foot of width to each of the four sides (adding

2

feet total to both overall width and overall length).

2

Write the area equation for the combined area.

(w + 2)(2w - 1) = 117

The total area of the garden and path combined is given as

117

square feet.

3

Expand and rearrange the equation into standard quadratic form

aw^2 + bw + c = 0

.

2w^2 + 3w - 119 = 0

Expanding

(w + 2)(2w - 1)

gives

2w^2 + 3w - 2

. Subtracting

117

from both sides yields the standard form.

4

Factor the quadratic equation over the integers.

(2w + 17)(w - 7) = 0

We find two numbers that multiply to

2 \times (-119) = -238

and sum to

3

. These numbers are

17

and

-14

. Rewriting the middle term and factoring by grouping yields

(2w + 17)(w - 7) = 0

.

5

Solve for

w

and select the mathematically and physically valid solution.

w = 7

(discarding the negative root

w = -8.5

)

A physical measurement like width must be positive.

Key Concept

Solving quadratic word problems by setting up a quadratic equation and solving it by factoring.

Estimated Time:2m 0s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution