All practice questions

Question 1Question →

The quadratic expression $x^2 - 5x - 14$ can be factored completely into $(x + a)(x + b)$ , where $a$ and $b$ are integers and $a > b$ . What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

To factor

x^2 - 5x - 14

, we look for two integers that multiply to

-14

and add to

-5

. These integers are

-7

and

2

. Thus, the factored form is

(x - 7)(x + 2)

, which corresponds to

(x + a)(x + b)

where the two constant values are

-7

and

2

. Given the condition

a > b

, the larger value must be assigned to

a

. Since

2 > -7

, we find

a = 2

.

Step-by-Step Solution

1

Find two integers that multiply to the constant term

-14

and add to the linear coefficient

-5

.

The two integers are

-7

and

2

.

Since

(-7) \times 2 = -14

and

-7 + 2 = -5

, these integers satisfy the requirements for factoring the quadratic trinomial.

2

Write the quadratic expression in its factored form

(x + p)(x + q)

.

(x - 7)(x + 2)

The quadratic expression

x^2 + Bx + C

factors into

(x + p)(x + q)

where

p

and

q

are the found integers.

3

Compare the factored form to the template

(x + a)(x + b)

under the condition

a > b

.

The set of constants is

\{-7, 2\}

. Since

2 > -7

, we assign

a = 2

and

b = -7

.

This satisfies the requirement that the integer

a

is strictly greater than the integer

b

.

Key Concept

Factoring quadratic trinomials with a leading coefficient of 1

Estimated Time:45s

Question 2Question →

The polynomial $x^2 + 8x + 15$ can be factored into the form $(x + a)(x + b)$ , where $a$ and $b$ are integers such that $a < b$ . What is the value of $2a + b$ ?

Show answer & explanation

Answer: 11

Answer

The value of

2a + b

is 11.

Factoring the trinomial

x^2 + 8x + 15

gives

(x + 3)(x + 5)

. Since

a < b

, we must have

a = 3

and

b = 5

. Thus,

2a + b = 2(3) + 5 = 11

.

Step-by-Step Solution

1

Factor the quadratic expression

x^2 + 8x + 15

.

(x + 3)(x + 5)

To factor the trinomial, we find two integers that multiply to the constant term 15 and add to the linear coefficient 8. The integers 3 and 5 satisfy these requirements.

2

Assign the values to

a

and

b

under the condition

a < b

.

a = 3

and

b = 5

Comparing

(x + 3)(x + 5)

to

(x + a)(x + b)

gives the values 3 and 5. The condition

a < b

dictates that the smaller value 3 goes to

a

and the larger value 5 goes to

b

.

3

Calculate the value of

2a + b

.

11

Substitute

a = 3

and

b = 5

into the expression:

2(3) + 5 = 6 + 5 = 11

.

Key Concept

Factoring quadratic trinomials

Question 3Question →

If $3x(x - 3) = 2(x + 3) - 12$ , what is the product of the two solutions to this equation?

Show answer & explanation

Answer: 2

Answer

The product of the two solutions to the equation is 2.

The correct answer is 2. Expanding both sides of the equation

3x(x - 3) = 2(x + 3) - 12

gives

3x^2 - 9x = 2x + 6 - 12

, which simplifies to

3x^2 - 9x = 2x - 6

. Subtracting

2x - 6

from both sides results in the standard form quadratic equation

3x^2 - 11x + 6 = 0

. Factoring this expression gives

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

. Setting each factor to zero yields the solutions

x = \frac{2}{3}

and

x = 3

. Multiplying these solutions gives

\frac{2}{3} \times 3 = 2

. Alternatively, by Vieta's formulas, the product of the roots of a quadratic equation

ax^2 + bx + c = 0

is

\frac{c}{a}

, which directly gives

\frac{6}{3} = 2

.

Step-by-Step Solution

1

Expand the expressions on both sides of the equation.

3x^2 - 9x = 2x + 6 - 12

, which simplifies to

3x^2 - 9x = 2x - 6

To prepare the equation for rearrangement into the standard quadratic form.

2

Move all terms to the left side of the equation to set it equal to zero.

3x^2 - 11x + 6 = 0

To write the quadratic equation in the standard form

ax^2 + bx + c = 0

.

3

Factor the quadratic trinomial by grouping.

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

To break down the quadratic equation into linear factors that can be solved individually.

4

Set each linear factor to zero and solve for

x

.

x = \frac{2}{3}

and

x = 3

According to the zero product property, if a product of factors is zero, at least one factor must be zero.

5

Calculate the product of the two solutions.

\frac{2}{3} \times 3 = 2

To find the product of the solutions as requested by the question.

Key Concept

Solving quadratic equations by factoring after expanding and rearranging terms

Question 4Question →

A charity walkathon organizer pledges to donate a base amount of $\$150$ plus $\$2.50$ for every kilometer completed by each participant. On the day of the event, a participant completes a distance that is $3$ kilometers less than twice their training average distance. If the organizer's donation for this participant is $\$212.50$ , what is the participant's training average distance, in kilometers?

Show answer & explanation

Answer: 14

Answer

The participant's training average distance is

14

kilometers.

The correct answer is

14

. To find this value, we write the donation relationship as

150 + 2.50d = 212.50

, where

d

is the distance completed. Solving for

d

gives

d = 25

kilometers. Next, we translate the phrase '3 kilometers less than twice their training average distance' into the expression

2A - 3

, where

A

is the training average. Equating this to the completed distance gives

2A - 3 = 25

. Solving for

A

yields

A = 14

.

Step-by-Step Solution

1

Set up an equation for the total donation as a function of the distance completed,

d

, in kilometers.

150 + 2.50d = 212.50

The total donation is the sum of the flat base donation of

\$150

and the rate of

\$2.50

per kilometer completed.

2

Solve the equation for the completed distance,

d

.

d = 25

Subtracting

150

from both sides of the equation yields

2.50d = 62.50

. Dividing both sides by

2.50

yields

d = 25

.

3

Translate the relationship between the completed distance,

d

, and the training average distance,

A

, into an equation.

2A - 3 = 25

The phrase '3 kilometers less than twice their training average distance' translates mathematically to

2A - 3

. Since the completed distance is

25

kilometers, we set the expression equal to

25

.

4

Solve the equation for the training average distance,

A

.

A = 14

Adding

3

to both sides of the equation yields

2A = 28

. Dividing both sides by

2

yields

A = 14

.

Key Concept

Translating word problems into multi-step linear equations and solving for the unknown variable.

Estimated Time:1m 30s

Question 5Question →

Line $p$ is defined by the equation $2x + 5y = 10$ . Line $q$ is perpendicular to line $p$ and has a $y$ -intercept of $-3$ . If line $q$ passes through the point $(a, 7)$ , what is the value of $a$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a

is 4.

First, the equation of line

p

,

2x + 5y = 10

, is converted to slope-intercept form to find its slope:

y = -\frac{2}{5}x + 2

. This indicates the slope of line

p

is

-\frac{2}{5}

. Because line

q

is perpendicular to line

p

, its slope must be the negative reciprocal of

-\frac{2}{5}

, which is

\frac{5}{2}

. Using the given

y

-intercept of

-3

, the equation of line

q

is written as

y = \frac{5}{2}x - 3

. Substituting the coordinates of the point

(a, 7)

into this equation gives

7 = \frac{5}{2}a - 3

. Adding

3

to both sides results in

10 = \frac{5}{2}a

, and solving for

a

yields

a = 4

.

Step-by-Step Solution

1

Find the slope of line

p

.

The slope of line

p

is

-\frac{2}{5}

.

Rewriting the equation

2x + 5y = 10

in slope-intercept form (

y = mx + b

) gives

5y = -2x + 10

, which simplifies to

y = -\frac{2}{5}x + 2

. The slope

m

is the coefficient of

x

, which is

-\frac{2}{5}

.

2

Determine the slope of line

q

.

The slope of line

q

is

\frac{5}{2}

.

Since line

q

is perpendicular to line

p

, its slope is the negative reciprocal of line

p

's slope:

-\frac{1}{-\frac{2}{5}} = \frac{5}{2}

.

3

Write the equation of line

q

.

The equation of line

q

is

y = \frac{5}{2}x - 3

.

Line

q

has a slope of

\frac{5}{2}

and a

y

-intercept of

-3

. Using the slope-intercept form

y = mx + b

gives

y = \frac{5}{2}x - 3

.

4

Substitute the point

(a, 7)

into the equation of line

q

and solve for

a

.

a = 4

Substituting

x = a

and

y = 7

into the equation gives

7 = \frac{5}{2}a - 3

. Adding

3

to both sides yields

10 = \frac{5}{2}a

. Multiplying both sides by

2

gives

20 = 5a

, and dividing by

5

results in

a = 4

.

Key Concept

The slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope.

Estimated Time:1m 30s

Question 6Question →

A library display shelf has space for $5$ distinct books aligned in a row. The librarian has $3$ different science fiction books and $2$ different biography books. How many different row arrangements of these $5$ books are possible if the $2$ biography books must not be placed next to each other?

Show answer & explanation

Answer: 72

Answer

72 different row arrangements are possible.

To find the number of arrangements where the

2

biography books are not adjacent, subtract the number of arrangements where they are adjacent (

4! \times 2! = 48

) from the total possible arrangements (

5! = 120

), resulting in

120 - 48 = 72

.

Step-by-Step Solution

1

Calculate the total number of unrestricted arrangements of all 5 distinct books.

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

There are

5

distinct books, so there are

5!

total ways to arrange them in a line.

2

Calculate the number of arrangements in which the 2 biography books are placed next to each other.

4! \times 2! = 24 \times 2 = 48

Treat the

2

biography books as a single combined unit. This leaves

4

units to arrange (

3

science fiction books plus

1

biography unit), which can be ordered in

4! = 24

ways. Inside the unit, the

2

biography books can be arranged in

2! = 2

ways.

3

Subtract the number of adjacent biography arrangements from the total number of arrangements.

120 - 48 = 72

Subtracting the restricted outcomes (biography books adjacent) from the total possible outcomes gives the number of valid arrangements (biography books separated).

Key Concept

Permutations and Counting Methods with Complementary Counting

Question 7Question →

If $x = -2$ , $y = 5$ , and $z = -\frac{1}{2}$ , what is the value of the algebraic expression $\frac{x^3 y + z^{-2}}{y - 2x}$ ?

Show answer & explanation

Answer: -4

Answer

The value of the expression is -4.

Substituting

x = -2

,

y = 5

, and

z = -\frac{1}{2}

into the expression yields

x^3 y = (-2)^3(5) = -40

and

z^{-2} = \left(-\frac{1}{2}\right)^{-2} = 4

, making the numerator

-40 + 4 = -36

. Evaluating the denominator gives

y - 2x = 5 - 2(-2) = 9

. Dividing

-36

by

9

results in

-4

.

Step-by-Step Solution

1

Evaluate the terms in the numerator individually.

x^3 y = (-2)^3(5) = -40

and

z^{-2} = \left(-\frac{1}{2}\right)^{-2} = 4

.

Negative bases raised to odd powers retain a negative sign, while negative exponents represent the reciprocal raised to a positive power.

2

Calculate the total numerator value.

-40 + 4 = -36

.

Summing the two evaluated terms gives the complete numerator.

3

Evaluate the denominator expression.

y - 2x = 5 - 2(-2) = 5 + 4 = 9

.

Subtracting a negative quantity is equivalent to adding its positive counterpart.

4

Divide the numerator by the denominator.

\frac{-36}{9} = -4.

Dividing a negative integer by a positive integer yields a negative quotient.

Key Concept

Evaluating Algebraic Expressions with Negative Integers and Negative Exponents

Question 8Question →

What is the value of the expression $\frac{-2^4 - 6 \times |2 - 2 \times 3|}{2 - (-2)^3}$ ?

Show answer & explanation

Answer: -4

Answer

The value of the expression is

-4

.

Evaluating the expression step-by-step using order of operations (PEMDAS) yields

-4

. First, simplifying the absolute value

|2 - 2 \times 3|

gives

4

. Next, the exponent

-2^4

evaluates to

-16

, and the exponent

(-2)^3

in the denominator evaluates to

-8

. The numerator simplifies to

-16 - 6(4) = -40

, and the denominator simplifies to

2 - (-8) = 10

. Finally, division gives

-40 \div 10 = -4

.

Step-by-Step Solution

1

Simplify the absolute value expression in the numerator

|2 - 2 \times 3| = |-4| = 4

By the order of operations, multiplication must be performed before subtraction inside the grouping symbols (absolute value).

2

Evaluate the exponential terms in the numerator and denominator

-2^4 = -16

and

(-2)^3 = -8

In the term

-2^4

, the negative sign is not grouped with the base, so it evaluates to

-(2 \times 2 \times 2 \times 2) = -16

. In

(-2)^3

, the base is negative, evaluating to

(-2) \times (-2) \times (-2) = -8

.

3

Calculate the final values of the numerator and the denominator

Numerator =

-40

, Denominator =

10

For the numerator, perform the multiplication before the subtraction:

-16 - (6 \times 4) = -16 - 24 = -40

. For the denominator, subtract the negative number:

2 - (-8) = 2 + 8 = 10

.

4

Divide the numerator by the denominator

-4

The quotient of

-40

and

10

is

-4

.

Key Concept

Order of operations (PEMDAS) containing exponents, absolute values, and signed numbers

Estimated Time:1m 15s

Question 9Question →

If the product of the complex numbers $4 - 2i$ and $k + 6i$ is a real number, where $k$ is a real constant and $i = \sqrt{-1}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

12

The product of the complex numbers

(4 - 2i)(k + 6i)

expands to

4k + 24i - 2ki - 12i^2

. Substituting

i^2 = -1

simplifies the expression to

(4k + 12) + (24 - 2k)i

. For this expression to represent a real number, the imaginary part must be zero:

24 - 2k = 0

, which yields

k = 12

.

Step-by-Step Solution

1

Multiply the two complex numbers

(4 - 2i)

and

(k + 6i)

using the FOIL method.

4k + 24i - 2ki - 12i^2

To find the product of the two complex expressions.

2

Substitute

i^2 = -1

into the expression and group the real and imaginary parts.

(4k + 12) + (24 - 2k)i

To simplify the expression into standard complex form

a + bi

.

3

Set the imaginary part of the resulting complex number to

0

and solve for

k

.

24 - 2k = 0 \implies k = 12

A complex number is real if and only if its imaginary part is equal to zero.

Key Concept

Complex multiplication and the definition of a real number in the complex plane

Question 10Question →

For all non-zero real numbers $a$ and $b$ , the expression $(a^2 b^k)^3$ is equivalent to $a^6 b^{15}$ . What is the value of the integer $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of the integer

k

is 5.

To find the value of

k

, we simplify the expression

(a^2 b^k)^3

using exponent rules. According to the power of a product property,

(xy)^z = x^z y^z

, so

(a^2 b^k)^3 = (a^2)^3 (b^k)^3

. Next, applying the power of a power property,

(x^y)^z = x^{yz}

, we get

a^{2 \cdot 3} b^{k \cdot 3} = a^6 b^{3k}

. Since this expression is equivalent to

a^6 b^{15}

, we set the exponents of

b

equal to each other:

3k = 15

. Dividing both sides by 3 yields

k = 5

.

Step-by-Step Solution

1

Apply the power of a product rule to the expression

(a^2 b^k)^3

.

(a^2)^3 \cdot (b^k)^3

The power of a product rule states that

(xy)^z = x^z y^z

.

2

Apply the power of a power rule to simplify the exponents.

a^6 b^{3k}

The power of a power rule states that

(x^y)^z = x^{y \cdot z}

, so

(a^2)^3 = a^{2 \cdot 3} = a^6

and

(b^k)^3 = b^{3k}

.

3

Set the exponent of

b

in

a^6 b^{3k}

equal to the exponent of

b

in the equivalent expression

a^6 b^{15}

.

3k = 15

Since the expressions are equivalent for all non-zero real numbers, the exponents of like bases must be equal.

4

Solve the linear equation for

k

.

k = 5

Dividing both sides of

3k = 15

by 3 isolates the variable

k

.

Key Concept

Properties of exponents, specifically the power of a product rule

(xy)^z = x^z y^z

and the power of a power rule

(x^y)^z = x^{y \cdot z}

.

Estimated Time:45s

Question 11Question →

What is the value of $x$ that satisfies the equation $\frac{3}{4}(x - 8) = \frac{1}{2}x + 2$ ?

Show answer & explanation

Answer: 32

Answer

The correct value of

x

is

32

.

Distributing

\frac{3}{4}

on the left side yields

\frac{3}{4}x - 6

. Subtracting

\frac{1}{2}x

(which is

\frac{2}{4}x

) from both sides gives

\frac{1}{4}x - 6 = 2

. Adding 6 to both sides gives

\frac{1}{4}x = 8

, and multiplying by 4 yields the correct value of

32

.

Step-by-Step Solution

1

Distribute the fraction

\frac{3}{4}

to both terms inside the parentheses.

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

To simplify the equation by removing the parentheses.

2

Subtract

\frac{1}{2}x

from both sides of the equation.

\frac{1}{4}x - 6 = 2

To collect all terms with the variable

x

on one side of the equation.

3

Add 6 to both sides of the equation.

\frac{1}{4}x = 8

To isolate the term containing the variable.

4

Multiply both sides of the equation by 4.

x = 32

To solve for

x

by eliminating the coefficient of

\frac{1}{4}

.

Key Concept

Solving linear equations with variables on both sides and fractional coefficients

Alternative Method

Multiply the entire equation by the least common denominator of the fractions, which is 4, to clear the fractions before solving:

4 \cdot [\frac{3}{4}(x - 8)] = 4 \cdot [\frac{1}{2}x + 2] \implies 3(x - 8) = 2x + 8

. Then distribute:

3x - 24 = 2x + 8

. Subtract

2x

from both sides:

x - 24 = 8

. Add 24 to both sides:

x = 32

.

Estimated Time:45s

Question 12Question →

If the expression $\frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3}$ is simplified to the form $p^x q^y$ for all non-zero real numbers $p$ and $q$ , what is the value of $2x - y$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Applying the power of a power rule to the numerator yields

(p^3 q^{-2})^{-1} = p^{-3} q^2

. Applying the same rule to the denominator yields

(p^{-1} q^2)^3 = p^{-3} q^6

. Dividing the terms by subtracting exponents gives

p^{-3 - (-3)} q^{2-6} = p^0 q^{-4}

. Thus,

x = 0

and

y = -4

. Evaluating

2x - y

gives

2(0) - (-4) = 4

.

Step-by-Step Solution

1

Simplify the numerator using the power of a power property, which states that

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

.

(p^3 q^{-2})^{-1} = p^{3 \cdot (-1)} q^{-2 \cdot (-1)} = p^{-3} q^2

This distributes the exponent of

-1

to both factors inside the parentheses in the numerator.

2

Simplify the denominator using the power of a power property, which states that

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

.

(p^{-1} q^2)^3 = p^{-1 \cdot 3} q^{2 \cdot 3} = p^{-3} q^6

This distributes the exponent of

3

to both factors inside the parentheses in the denominator.

3

Combine the simplified numerator and denominator using the quotient property of exponents, which states that

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

.

\frac{p^{-3} q^2}{p^{-3} q^6} = p^{-3 - (-3)} q^{2 - 6} = p^0 q^{-4}

This simplifies division by subtracting the exponent in the denominator from the exponent in the numerator for each base.

4

Identify the values of

x

and

y

from the simplified form

p^0 q^{-4}

, and evaluate the final expression

2x - y

.

x = 0

and

y = -4

, so

2(0) - (-4) = 4

This substitutes the values of the exponents into the target algebraic expression to find the final numerical answer.

Key Concept

Properties of exponents including power of a power and quotient properties

Alternative Method

Alternatively, you can write the terms with positive exponents first:

\frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3} = \frac{(p^{-1} q^2)^3}{(p^3 q^{-2})^1} = \frac{p^{-3} q^6}{p^3 q^{-2}} = p^{-3-3} q^{6-(-2)} = p^{-6} q^8

, but this expression is equivalent to the original expression only if inverted properly. Direct distribution is less prone to inversion errors.

Estimated Time:1m 30s

Question 13Question →

What is the sum of the squares of the two real solutions to the quadratic equation $\frac{1}{2}x^2 - 3x + 2 = 0$ ?

Show answer & explanation

Answer: 28

Answer

The sum of the squares of the two real solutions is 28.

Multiplying the equation

\frac{1}{2}x^2 - 3x + 2 = 0

by 2 yields

x^2 - 6x + 4 = 0

. By Vieta's formulas, the sum of the solutions is

x_1 + x_2 = 6

and the product of the solutions is

x_1 x_2 = 4

. Using the algebraic identity

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

, the sum of the squares is

6^2 - 2(4) = 36 - 8 = 28

. Alternatively, using the quadratic formula on the simplified equation gives solutions of

3 + \sqrt{5}

and

3 - \sqrt{5}

. Squaring these values yields

14 + 6\sqrt{5}

and

14 - 6\sqrt{5}

, which sum to 28.

Step-by-Step Solution

1

Multiply the quadratic equation by 2 to clear the fraction.

x^2 - 6x + 4 = 0

Simplifying the fractional coefficients makes the equation easier to analyze and solve.

2

Identify the sum and product of the roots using Vieta's formulas.

x_1 + x_2 = 6

and

x_1 x_2 = 4

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-b/a

and the product is

c/a

.

3

Express the sum of the squares of the roots using the algebraic identity.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

This identity allows the direct calculation of the sum of squares without needing to find the individual roots.

4

Substitute the values of the sum and product into the identity.

6^2 - 2(4) = 36 - 8 = 28

Plugging in the sum of 6 and product of 4 yields the final numerical answer.

Key Concept

Vieta's formulas state that for a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

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

and the product of the roots is

x_1 x_2 = \frac{c}{a}

. Symmetric functions of roots like

x_1^2 + x_2^2

can be expressed in terms of these values.

Alternative Method

Find the roots of the equation directly using the quadratic formula. After simplifying to

x^2 - 6x + 4 = 0

, the roots are

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(4)}}{2} = 3 \pm \sqrt{5}

. Squaring both solutions gives

(3 + \sqrt{5})^2 = 14 + 6\sqrt{5}

and

(3 - \sqrt{5})^2 = 14 - 6\sqrt{5}

. Adding these squares together yields

(14 + 6\sqrt{5}) + (14 - 6\sqrt{5}) = 28

.

Estimated Time:1m 30s

Question 14Question →

If $a$ , $b$ , and $c$ are positive real numbers such that $a(b + c) = 20$ , $b(c + a) = 13$ , and $c(a + b) = 25$ , what is the value of the product $abc$ ?

Show answer & explanation

Answer: 24

Answer

The product

abc

is equal to 24.

By applying the distributive property, we expand the system to

ab + ac = 20

,

ab + bc = 13

, and

bc + ac = 25

. Adding these yields

2(ab + bc + ac) = 58

, so

ab + bc + ac = 29

. We isolate the individual products:

bc = 9

,

ac = 16

, and

ab = 4

. Multiplying these gives

(abc)^2 = 4 \times 9 \times 16 = 576

. Since the variables are positive,

abc = \sqrt{576} = 24

.

Step-by-Step Solution

1

Expand the equations using the distributive property.

ab + ac = 20

,

ab + bc = 13

, and

bc + ac = 25

This allows us to work with the pairwise products

ab

,

bc

, and

ac

directly.

2

Sum the three equations and divide by 2.

ab + bc + ac = 29

By adding the equations, each pairwise product appears twice:

(ab + ac) + (ab + bc) + (bc + ac) = 2(ab + bc + ac) = 58

. Dividing by 2 gives their sum.

3

Solve for each pairwise product by subtracting the original equations from the sum.

bc = 9

,

ac = 16

, and

ab = 4

Subtracting

ab + ac = 20

from

ab + bc + ac = 29

isolates

bc = 9

. Similarly, subtracting

ab + bc = 13

isolates

ac = 16

, and subtracting

bc + ac = 25

isolates

ab = 4

.

4

Multiply the pairwise products and take the square root.

(abc)^2 = 576 \implies abc = 24

Multiplying

(ab)(bc)(ac)

gives

(abc)^2

. Since

a

,

b

, and

c

are positive, their product

abc

must also be positive, so we take the positive square root of

576

, which is

24

.

Key Concept

Distributive property of multiplication over addition, and properties of equality in systems of equations.

Question 15Question →

What is the value of the expression $\frac{3 \times 4^2 - 2 \times (5 - 8)^2}{2 - (-2)^3}$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is 3.

Following the standard order of operations (PEMDAS), we first calculate the expression inside the parentheses:

5 - 8 = -3

. We then evaluate the exponents:

4^2 = 16

,

(-3)^2 = 9

, and

(-2)^3 = -8

. Substituting these back into the expression, the numerator becomes

3(16) - 2(9) = 48 - 18 = 30

, and the denominator becomes

2 - (-8) = 2 + 8 = 10

. Dividing the numerator by the denominator gives

\frac{30}{10} = 3

.

Step-by-Step Solution

1

Simplify the grouping inside the parentheses

5 - 8 = -3

According to the order of operations, terms inside grouping symbols must be evaluated first.

2

Evaluate the exponential terms

4^2 = 16

,

(-3)^2 = 9

, and

(-2)^3 = -8

Exponents are evaluated next after parentheses.

3

Perform multiplication in the numerator

3 \times 16 = 48

and

2 \times 9 = 18

Multiplication has priority over subtraction.

4

Simplify the numerator and the denominator by performing subtraction and addition

Numerator:

48 - 18 = 30

; Denominator:

2 - (-8) = 10

Addition and subtraction are performed next from left to right.

5

Divide the simplified numerator by the simplified denominator

\frac{30}{10} = 3

Perform the final division to find the value of the fraction.

Key Concept

Order of Operations

Question 16Question →

What is the value of the expression $\frac{18 - 3 \times 4}{|-2| + 1}$ ?

Show answer & explanation

Answer: 2

Answer

The value of the expression is 2.

To evaluate the expression, we simplify the numerator and denominator separately first. In the numerator, we perform the multiplication before the subtraction:

18 - 3 \times 4 = 18 - 12 = 6

. In the denominator, we evaluate the absolute value and then add:

|-2| + 1 = 2 + 1 = 3

. Finally, we divide the simplified numerator by the simplified denominator:

\frac{6}{3} = 2

.

Step-by-Step Solution

1

Evaluate the multiplication in the numerator.

3 \times 4 = 12

, leaving the numerator as

18 - 12

.

According to the order of operations, multiplication must be performed before subtraction.

2

Subtract the values in the numerator.

18 - 12 = 6

.

Simplify the numerator expression completely.

3

Evaluate the absolute value in the denominator.

|-2| = 2

, leaving the denominator as

2 + 1

.

Absolute value represents the non-negative distance from zero, so

|-2| = 2

.

4

Add the values in the denominator.

2 + 1 = 3

.

Simplify the denominator expression completely.

5

Divide the numerator by the denominator.

\frac{6}{3} = 2

.

Compute the final fraction value.

Key Concept

Evaluating expressions using the order of operations (PEMDAS) and absolute value properties.

Question 17Question →

What is the greatest common factor of $36$ and $54$ ?

Show answer & explanation

Answer: 18

Answer

The greatest common factor of

36

and

54

is

18

.

The greatest common factor of

36

and

54

is

18

because it is the largest integer that divides both numbers evenly (

36 \div 18 = 2

and

54 \div 18 = 3

).

Step-by-Step Solution

1

Find the prime factorization of

36

and

54

.

36 = 2^2 \times 3^2

and

54 = 2 \times 3^3

Prime factorization decomposes each number into its basic prime building blocks.

2

Identify the common prime factors with their lowest exponents.

The common prime factors are

2

with an exponent of

1

(

2^1

) and

3

with an exponent of

2

(

3^2

).

The greatest common factor consists of the product of the lowest powers of the shared prime factors.

3

Multiply the common factors raised to their lowest exponents.

2^1 \times 3^2 = 2 \times 9 = 18

Calculating this product gives the greatest common factor.

Key Concept

The greatest common factor (GCF) of two integers is the largest integer that divides both numbers without leaving a remainder. It is calculated by taking the product of the lowest powers of all common prime factors.

Alternative Method

List all factors of both numbers: Factors of

36

are

1, 2, 3, 4, 6, 9, 12, 18, 36

. Factors of

54

are

1, 2, 3, 6, 9, 18, 27, 54

. The largest number present in both lists is

18

.

Estimated Time:45s

Question 18Question →

What is the value of the expression $(-2)^4 - 3 \times \frac{|12 - 18|}{-2} + 5 \times (3 - 5)^3$ ?

Show answer & explanation

Answer: -15

Answer

The correct answer is -15.

Evaluating the expression according to order of operations yields -15. First, simplify inside the grouping symbols to obtain

(3 - 5) = -2

and

|12 - 18| = 6

. Next, evaluate exponents to obtain

(-2)^4 = 16

and

(-2)^3 = -8

. After this, multiply and divide from left to right to obtain

-3 \times \frac{6}{-2} = 9

and

5 \times (-8) = -40

. Finally, add and subtract from left to right to obtain

16 + 9 - 40 = -15

.

Step-by-Step Solution

1

Evaluate expressions inside grouping symbols.

The grouping symbols evaluate to

(3 - 5) = -2

and

|12 - 18| = |-6| = 6

.

Operations inside parentheses and absolute value bars are prioritized first in the order of operations.

2

Evaluate the exponential terms.

(-2)^4 = 16

and

(-2)^3 = -8

.

Exponents must be calculated after grouping symbols and before multiplication or division.

3

Evaluate multiplication and division from left to right.

-3 \times \frac{6}{-2} = 9

and

5 \times (-8) = -40

.

Multiplication and division are equal in precedence and must be executed in order from left to right.

4

Perform addition and subtraction from left to right.

16 + 9 - 40 = -15

.

Addition and subtraction are the final operations performed, in order from left to right.

Key Concept

Order of operations with exponents, absolute values, and signed numbers.

Question 19Question →

What is the value of the expression $3 \times (4^2 - 10) - 5 \times |-2|$ ?

Show answer & explanation

Answer: 8

Answer

The correct value of the expression is 8.

Evaluating the expression following the order of operations (parentheses, exponents, multiplication/division, and addition/subtraction) results in 8.

Step-by-Step Solution

1

Evaluate the exponent inside the parentheses

4^2 = 16

Exponents must be evaluated before basic arithmetic operations inside parentheses.

2

Subtract the numbers within the parentheses

16 - 10 = 6

Operations enclosed in parentheses are prioritized first.

3

Evaluate the absolute value term

|-2| = 2

The absolute value of a negative number is its positive distance from zero.

4

Perform the multiplications from left to right

3 \times 6 = 18

and

5 \times 2 = 10

Multiplications are performed before addition and subtraction.

5

Perform the final subtraction

18 - 10 = 8

Subtraction is performed last according to the order of operations.

Key Concept

Order of Operations and Number Properties

Question 20Question →

What is the sum of the distinct prime factors of $30$ ?

Show answer & explanation

Answer: 10

Answer

The sum of the distinct prime factors of

30

is

10

.

The prime factorization of

30

is

2 \times 3 \times 5

. The distinct prime factors are

2

,

3

, and

5

, and their sum is

10

.

Step-by-Step Solution

1

Find the prime factorization of

30

.

30 = 2 \times 3 \times 5

To break the composite number

30

down into its prime components.

2

Identify the distinct prime factors.

The distinct prime factors are

2

,

3

, and

5

.

Only prime numbers that divide

30

should be included in the sum.

3

Add the distinct prime factors.

2 + 3 + 5 = 10

To calculate the final sum as requested by the question.

Key Concept

Prime factorization is the process of factoring a composite number into a product of prime numbers. The distinct prime factors of a number are the unique prime numbers that divide it.

Answer

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