All practice questions

For the function $h(x) = 5 - 3x$ , what is the value of $h(h(2))$ ?

Show answer & explanation

Answer: 8

Answer

8

To find

h(h(2))

, we first calculate the value of the inner function,

h(2) = 5 - 3(2) = 5 - 6 = -1

. We then substitute this result back into the function to evaluate the outer function:

h(-1) = 5 - 3(-1) = 5 + 3 = 8

. Therefore, the correct value is 8.

Step-by-Step Solution

1

Evaluate the inner function

h(2)

h(2) = -1

To evaluate a nested function composition of the form

h(h(x))

, first calculate the value of the inner function at the given input.

2

Evaluate the outer function

h(-1)

using the result from Step 1

h(h(2)) = h(-1) = 8

Substitute the output of the inner function,

-1

, as the new input for the outer function

h(x)

.

Key Concept

Evaluating the composition of a function with itself

Question 1882Question →

A security passcode consists of a sequence of 4 digits chosen from the digits 0 through 9. To be valid, a passcode must contain at least one repeated digit, but no digit can appear more than twice. Additionally, the passcode cannot end with an odd digit. How many different valid security passcodes can be created?

Show answer & explanation

Answer: 2295

Answer

There are 2,295 different valid security passcodes.

The correct answer of 2,295 is found by subtracting all invalid passcodes from the total possible passcodes. The total number of 4-digit passcodes ending in an even digit is 5,000. The invalid passcodes are those with all distinct digits (2,520), those with a digit repeated three times (180), and those with all four digits identical (5). Subtracting these gives 5,000 - 2,520 - 180 - 5 = 2,295.

Step-by-Step Solution

1

Calculate the total number of 4-digit passcodes ending in an even digit.

5,000

The last digit must be even (0, 2, 4, 6, or 8) to not be odd, giving 5 choices. The first three digits can be any of the 10 digits from 0 through 9. By the Fundamental Counting Principle, the total number of passcodes is 10 * 10 * 10 * 5 = 5,000.

2

Calculate the number of invalid passcodes where all 4 digits are distinct.

2,520

For all 4 digits to be distinct, the last digit must be chosen from the 5 even digits. The remaining 3 positions must be filled with 3 distinct digits chosen from the remaining 9 digits. There are 9 * 8 * 7 = 504 ways to choose these. This gives 504 * 5 = 2,520 passcodes.

3

Calculate the number of invalid passcodes where a single digit is repeated three times.

180

If the repeated digit is the last digit, there are 5 choices for that digit, and the single distinct digit (9 choices) can be placed in any of the 3 remaining positions, giving 5 * 9 * 3 = 135 passcodes. If the repeated digit is not the last digit, the three identical digits occupy the first three positions, and the last digit (5 choices) is distinct from them (9 choices), giving 5 * 9 = 45 passcodes. In total, 135 + 45 = 180 passcodes.

4

Calculate the number of invalid passcodes where all four digits are identical.

5

Since the last digit must be even, all four digits must be the same even digit (0000, 2222, 4444, 6666, or 8888), which gives 5 passcodes.

5

Subtract the invalid passcodes from the total number of passcodes.

2,295

Subtracting the passcodes with all distinct digits (2,520) and those with a digit repeated three or four times (180 + 5 = 185) from the total of 5,000 gives 5,000 - 2,520 - 185 = 2,295.

Key Concept

Complementary counting using permutations and partition analysis

Alternative Method

Instead of complementary counting, count the valid cases directly: Case A where the digit frequencies are [2, 1, 1] (which yields 2,160 codes) and Case B where the digit frequencies are [2, 2] (which yields 135 codes). Adding these yields 2,160 + 135 = 2,295 codes.

Estimated Time:3m 0s

Question 1883Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 12y + 27 = 0$ . A parabola that opens downward has its vertex at $(0, k)$ and is defined by the equation $y = -x^2 + k$ . If the system of equations consisting of this circle and parabola has exactly three distinct real solution points, what is the value of $k$ ?

Show answer & explanation

Answer: 9

Answer

The value of

k

is 9.

The correct value of

k

is 9 because when

k=9

, the system of equations reduces to a quadratic in

y

with roots

y=9

and

y=4

. Both roots satisfy the real-number constraint

y \leq 9

for the parabola

x^2 = 9-y

, producing three distinct real solutions:

(0, 9)

,

(\sqrt{5}, 4)

, and

(-\sqrt{5}, 4)

.

Step-by-Step Solution

1

Complete the square for the circle's equation.

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

To identify the circle's center at

(0, 6)

and radius

R=3

for geometric interpretation.

2

Express

x^2

in terms of

y

using the parabola's equation.

x^2 = k - y

To substitute into the circle's equation and eliminate the

x

variable.

3

Substitute

x^2

into the circle's equation and simplify.

y^2 - 13y + (k+27) = 0

To create a quadratic equation in

y

representing the y-coordinates of the intersection points.

4

Set

y = k

in the quadratic equation.

k^2 - 12k + 27 = 0

, which factors as

(k-3)(k-9) = 0

An intersection must lie on the y-axis (

x=0

, which means

y=k

) to yield an odd number of intersection points.

5

Verify which candidate value of

k

yields exactly three real solutions.

For

k=3

, the solutions are restricted because

y=10

gives no real

x

value, resulting in only 1 solution. For

k=9

, the roots

y=9

and

y=4

both yield real

x

values, resulting in exactly 3 solutions:

(0, 9)

,

(\sqrt{5}, 4)

, and

(-\sqrt{5}, 4)

.

The algebraic condition for real

x

coordinates is

x^2 = k - y \geq 0

, so we must verify that the roots

y

satisfy

y \leq k

.

Key Concept

Solving systems of non-linear equations algebraically and analyzing the number of real intersection points under coordinate constraints.

Alternative Method

Geometrically, a parabola opening downward with its vertex on the y-axis will intersect a circle centered on the y-axis in exactly three points if and only if its vertex is at the top of the circle and its curvature is less than that of the circle at that point. Completing the square for the circle

x^2 + y^2 - 12y + 27 = 0

gives

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

, which shows the top point of the circle is

(0, 9)

. Thus, the vertex of the downward-opening parabola must be at

(0, 9)

, meaning

k = 9

. We then algebraically verify that this curvature indeed allows two other real intersections.

Estimated Time:3m 0s

Question 1884Question →

For all real values of $x$ , what is the real solution to the equation $\sqrt{x+13} - 1 = x$ ?

-4

-3

3

4

11

Show answer & explanation

Answer: 3

Answer

The only real solution to the equation is 3.

Isolating the radical term gives

\sqrt{x+13} = x+1

. Squaring both sides yields

x+13 = x^2 + 2x + 1

. Rearranging into standard quadratic form gives

x^2 + x - 12 = 0

, which factors as

(x+4)(x-3) = 0

. This yields candidate solutions

x = 3

and

x = -4

. Checking these values in the original equation shows that

x = 3

is a valid solution because

\sqrt{3+13} - 1 = 3

, whereas

x = -4

is extraneous because

\sqrt{-4+13} - 1 = 2 \neq -4

.

Step-by-Step Solution

1

Isolate the radical expression on one side of the equation.

\sqrt{x+13} = x+1

This prepares the equation for squaring both sides to eliminate the radical.

2

Square both sides of the equation.

x+13 = (x+1)^2 \implies x+13 = x^2 + 2x + 1

Squaring both sides eliminates the radical and yields a polynomial equation.

3

Move all terms to one side to set the quadratic equation equal to zero.

x^2 + x - 12 = 0

This puts the equation in the standard quadratic form

ax^2 + bx + c = 0

so it can be solved by factoring.

4

Factor the quadratic equation.

(x+4)(x-3) = 0 \implies x = 3

or

x = -4

Factoring allows us to find the potential roots of the quadratic equation.

5

Check both candidate solutions in the original equation to identify any extraneous solutions.

For

x = 3

:

\sqrt{3+13} - 1 = 4 - 1 = 3

(valid). For

x = -4

:

\sqrt{-4+13} - 1 = 3 - 1 = 2 \neq -4

(extraneous).

Squaring both sides of an equation can introduce extraneous roots that must be checked and discarded.

Key Concept

Solving radical equations and checking for extraneous solutions

Estimated Time:1m 30s

Question 1885Question →

If $f(x) = 3x - 5$ and $g(x) = (x - 1)^2$ , what is the value of $f(g(4))$ ?

4

12

22

36

40

Show answer & explanation

Answer: 22

Answer

22

To find

f(g(4))

, we evaluate the inner function first:

g(4) = (4 - 1)^2 = 3^2 = 9

. We then evaluate the outer function at this result:

f(9) = 3(9) - 5 = 27 - 5 = 22

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = 4

g(4) = (4 - 1)^2 = 3^2 = 9

In function composition

f(g(x))

, the inner function must be evaluated first.

2

Substitute the result from the first step into the outer function

f(x)

f(9) = 3(9) - 5

The output of the inner function becomes the input of the outer function.

3

Calculate the final value of

f(9)

following the order of operations

f(9) = 27 - 5 = 22

Multiplication must be performed before subtraction.

Key Concept

Function Evaluation and Composition

Question 1886Question →

A food truck charges a flat fee of $6 for a custom lunch box container, plus$ 8 for each scoop of specialty salad added to the box. If a customer paid a total of $38 for one custom lunch box filled with specialty salad, how many scoops of specialty salad did they receive?

14

32

4

5

8

Show answer & explanation

Answer: 4

Answer

4 scoops of specialty salad

The correct answer is 4. The total cost is represented by the equation

6 + 8s = 38

, where

s

represents the number of scoops. Subtracting the flat container fee of

6 from the total of

38 leaves

32. Dividing

32 by the per-scoop cost of $8 yields 4 scoops.

Step-by-Step Solution

1

Set up the linear equation representing the total cost.

6 + 8s = 38

, where

s

is the number of scoops of specialty salad.

The total cost consists of a one-time flat fee of

6 plus

8 per scoop of salad.

2

Subtract the flat fee from both sides of the equation.

8s = 32

To isolate the variable term, we subtract the constant cost of the container from the total cost.

3

Divide both sides of the equation by the cost per scoop.

s = 4

Dividing the remaining cost by the price per scoop yields the number of scoops purchased.

Key Concept

Translating a verbal description of a linear cost scenario into a one-variable linear equation and solving it.

Question 1887Question →

If $\frac{3}{5}x - 7 = 8$ , what is the value of $2x - 3$ ?

15

25

44

47

\frac{1}{3}

Show answer & explanation

Answer: 47

Answer

47

To find the value of

2x - 3

, first solve the equation

\frac{3}{5}x - 7 = 8

for

x

. Adding 7 to both sides of the equation gives

\frac{3}{5}x = 15

. Multiplying both sides by the reciprocal

\frac{5}{3}

yields

x = 25

. Finally, substitute

25

for

x

in the expression

2x - 3

to get

2(25) - 3 = 50 - 3 = 47

.

Step-by-Step Solution

1

Add 7 to both sides of the equation

\frac{3}{5}x - 7 = 8

to isolate the variable term.

\frac{3}{5}x = 15

Adding 7 to both sides eliminates the constant on the left side of the equation.

2

Multiply both sides of the equation by

\frac{5}{3}

to solve for

x

.

x = 25

Multiplying by the reciprocal of

\frac{3}{5}

isolates

x

on the left side of the equation.

3

Substitute

x = 25

into the expression

2x - 3

.

2(25) - 3 = 50 - 3 = 47

The question asks for the value of the expression

2x - 3

rather than just the variable

x

.

Key Concept

Solving two-step linear equations involving fractions, and evaluating algebraic expressions.

Alternative Method

Instead of solving for

x

first, you can express the target expression in terms of

\frac{3}{5}x

. Specifically, note that

2x - 3 = \frac{10}{3}(\frac{3}{5}x) - 3

. Since

\frac{3}{5}x = 15

, substituting this directly gives

\frac{10}{3}(15) - 3 = 50 - 3 = 47

.

Estimated Time:45s

Question 1888Question →

Which of the following expressions is equivalent to the expression below?

(3x^3 - 2y^2)^2 - 2(x^3 - 4y^2)(4x^3 - 5y^2)

x^9 + 30x^3y^2 - 36y^4

x^6 - 54x^3y^2 + 44y^4

x^6 + 30x^3y^2 - 36y^4

x^6 + 42x^3y^2 - 36y^4

-7x^6 + 72x^3y^2 - 76y^4

Show answer & explanation

Answer: $x^6 + 30x^3y^2 - 36y^4$

Answer

The expression

x^6 + 30x^3y^2 - 36y^4

.

To find the correct expression, we first square the binomial to obtain

9x^6 - 12x^3y^2 + 4y^4

. Next, we multiply the two binomials

(x^3 - 4y^2)(4x^3 - 5y^2)

to get

4x^6 - 21x^3y^2 + 20y^4

, and distribute the factor of 2 to obtain

8x^6 - 42x^3y^2 + 40y^4

. Finally, we subtract this from the squared binomial by distributing the negative sign to all terms:

9x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4

. Combining like terms yields

x^6 + 30x^3y^2 - 36y^4

.

Step-by-Step Solution

1

Expand the squared binomial

(3x^3 - 2y^2)^2

.

9x^6 - 12x^3y^2 + 4y^4

Use the binomial squaring formula

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

, where

a = 3x^3

and

b = 2y^2

.

2

Multiply the two binomials

(x^3 - 4y^2)(4x^3 - 5y^2)

.

4x^6 - 21x^3y^2 + 20y^4

Apply the FOIL method to expand the product:

4x^6 - 5x^3y^2 - 16x^3y^2 + 20y^4

, and combine the like middle terms.

3

Multiply the resulting trinomial from Step 2 by the constant factor of 2.

8x^6 - 42x^3y^2 + 40y^4

Distribute the factor of 2 to each term of the simplified trinomial expression.

4

Subtract the expression in Step 3 from the expression in Step 1.

x^6 + 30x^3y^2 - 36y^4

Distribute the negative sign to all terms of the subtracted polynomial:

9x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4

, and combine the remaining like terms.

Key Concept

Operations on Polynomials

Question 1889Question →

Which of the following is the completely factored form of the expression $3x^3 - 12x^2 + 12x$ ?

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

3x^2(x - 2)^2

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

3x(x - 2)^2

3x(x + 2)^2

Show answer & explanation

Answer: $3x(x - 2)^2$

Answer

The completely factored form of the expression is

3x(x - 2)^2

.

To factor the expression

3x^3 - 12x^2 + 12x

completely, we first look for the greatest common factor (GCF) of the three terms. The GCF of

3x^3

,

-12x^2

, and

12x

is

3x

. Factoring out

3x

yields

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

. Next, we factor the quadratic trinomial inside the parentheses. The expression

x^2 - 4x + 4

is a perfect square trinomial that factors into

(x - 2)^2

. Putting it all together, the completely factored form is

3x(x - 2)^2

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial

3x^3 - 12x^2 + 12x

.

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

Each term in the polynomial is divisible by

3

, and the lowest power of

x

common to all terms is

x^1

. Thus, the GCF is

3x

.

2

Factor the remaining quadratic trinomial

x^2 - 4x + 4

inside the parentheses.

(x - 2)^2

The trinomial

x^2 - 4x + 4

is a perfect square trinomial matching the form

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

, where

a = x

and

b = 2

.

3

Combine the factored parts to write the final completely factored expression.

3x(x - 2)^2

Bringing together the GCF and the factored trinomial yields the simplest, fully factored form.

Key Concept

Factoring a polynomial completely by first extracting the greatest common factor (GCF) and then factoring the remaining perfect square trinomial.

Alternative Method

Instead of factoring directly, you can expand the answer choices to see which one is equivalent to the original polynomial. For example, expanding the correct option:

3x(x-2)^2 = 3x(x^2 - 4x + 4) = 3x^3 - 12x^2 + 12x

. This matches the original expression.

Estimated Time:1m 0s

Question 1890Question →

Let the complex number $z$ be defined as $z = (4 - 3i)(1 + 2i) + 5i^{14}$ , where $i = \sqrt{-1}$ . What is the real part of $z$ ?

Show answer & explanation

Answer: 5

Answer

The real part of the complex number

z

is

5

.

First, expand the product

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

to get

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

. Replacing

i^2

with

-1

gives

10 + 5i

. Next, simplify

5i^{14}

. Since

i^{14} = (i^4)^3 \cdot i^2 = 1^3 \cdot (-1) = -1

, the term becomes

-5

. Adding the components together gives

z = (10 + 5i) - 5 = 5 + 5i

. The real part of this complex number is

5

.

Step-by-Step Solution

1

Expand the product of the complex binomials

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

10 + 5i

Applying the distributive property gives

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

. Substituting

i^2 = -1

simplifies the expression to

4 + 5i + 6 = 10 + 5i

.

2

Simplify the power of the imaginary unit in

5i^{14}

-5

Since the powers of

i

cycle every 4 terms,

i^{14} = i^{12} \cdot i^2 = 1 \cdot (-1) = -1

. Therefore,

5i^{14} = 5(-1) = -5

.

3

Add the simplified terms together to find

z

5 + 5i

Adding the real and imaginary parts of the terms yields

z = (10 + 5i) + (-5) = 5 + 5i

.

4

Determine the real part of

z

5

A complex number is written in the form

a + bi

, where

a

represents the real part. For

5 + 5i

, the real part is

5

.

Key Concept

Complex multiplication and simplification of powers of the imaginary unit

Question 1891Question →

A certain value $x$ satisfies the relationship where the square of the difference between $x$ and $4$ is equal to $16$ decreased by $3$ times $x$ . If $x_1$ and $x_2$ are the two real solutions to this relationship, with $x_1 > x_2$ , what is the value of $3x_1 - 2x_2$ ?

3

6

10

15

33

Show answer & explanation

Answer: 15

Answer

The value of the expression is 15.

The value 15 is correct. First, translate the relationship into the algebraic equation

(x-4)^2 = 16 - 3x

. Expanding the left side yields

x^2 - 8x + 16 = 16 - 3x

. To solve the quadratic equation by factoring, rearrange the terms to set one side to zero: subtract 16 and add

3x

to both sides, which simplifies to

x^2 - 5x = 0

. Factoring the left-hand side gives

x(x - 5) = 0

. The solutions are

x = 5

and

x = 0

. Since the problem defines

x_1 > x_2

, the larger solution is

x_1 = 5

and the smaller solution is

x_2 = 0

. Substituting these values into the expression

3x_1 - 2x_2

yields

3(5) - 2(0) = 15

.

Step-by-Step Solution

1

Translate the verbal description into a mathematical equation.

(x - 4)^2 = 16 - 3x

The 'square of the difference between

x

and

4

' is represented as

(x-4)^2

, and '16 decreased by 3 times

x

' is represented as

16 - 3x

.

2

Expand the squared binomial on the left side of the equation.

x^2 - 8x + 16 = 16 - 3x

Using the binomial expansion formula

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

, expand

(x-4)^2

to

x^2 - 2(4)(x) + 4^2 = x^2 - 8x + 16

.

3

Rearrange the equation to set one side to zero.

x^2 - 5x = 0

Add

3x

and subtract

16

from both sides of the equation to gather all terms on one side:

x^2 - 8x + 3x + 16 - 16 = 0

, which simplifies to

x^2 - 5x = 0

.

4

Factor the quadratic equation.

x(x - 5) = 0

Factor out the greatest common factor, which is

x

.

5

Solve for the roots of the equation.

x = 0

or

x = 5

Set each factor to zero:

x = 0

or

x - 5 = 0

.

6

Identify the values of

x_1

and

x_2

and calculate the required expression.

x_1 = 5

,

x_2 = 0

, so

3x_1 - 2x_2 = 15

Since

x_1 > x_2

, assign

x_1 = 5

and

x_2 = 0

. Calculate

3(5) - 2(0) = 15

.

Key Concept

Solving quadratic equations by rearranging terms, expanding binomials, factoring out the greatest common factor, and solving for roots.

Question 1892Question →

When the polynomial $12x^2 + 11x - 15$ is factored completely into the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the constant term $d$ ?

Show answer & explanation

Answer: 5

Answer

The value of the constant term

d

is

5

.

Factoring the trinomial

12x^2 + 11x - 15

completely gives

(4x - 3)(3x + 5)

. Applying the constraint

a > c > 0

means the factor with the larger

x

-coefficient must be written first in the template

(ax + b)(cx + d)

. This yields

a = 4

,

b = -3

,

c = 3

, and

d = 5

. Thus, the constant term

d

is

5

.

Step-by-Step Solution

1

Find the factor pair for the AC method.

We need two numbers that multiply to

12 \times (-15) = -180

and add up to

11

. The numbers are

20

and

-9

.

This allows us to split the linear middle term to factor by grouping.

2

Rewrite the polynomial and factor by grouping.

12x^2 + 20x - 9x - 15 = 4x(3x + 5) - 3(3x + 5) = (4x - 3)(3x + 5)

.

Grouping the first two terms and the last two terms reveals a common binomial factor of

(3x + 5)

.

3

Apply the given inequality constraints to match the template.

Comparing

(4x - 3)(3x + 5)

to

(ax + b)(cx + d)

with

a > c > 0

yields

a = 4

,

b = -3

,

c = 3

, and

d = 5

.

Since the lead coefficient

4

is greater than

3

, the factor

(4x - 3)

must correspond to

(ax + b)

.

Key Concept

Factoring quadratic trinomials of the form

Ax^2 + Bx + C

using the grouping (AC) method.

Question 1893Question →

A chemist wants to create $100$ milliliters of a $42.5\%$ acid solution by mixing three different acid solutions: a $10\%$ acid solution, a $20\%$ acid solution, and an $80\%$ acid solution. She decides that the volume of the $20\%$ acid solution used must be exactly $3$ times the volume of the $10\%$ acid solution used. What is the difference, in milliliters, between the volume of the $80\%$ acid solution and the volume of the $10\%$ acid solution used in the final mixture?

Show answer & explanation

Answer: 25

Answer

The difference between the volume of the

80\%

acid solution and the volume of the

10\%

acid solution is

25

milliliters.

Solving the system of equations yields that

15

milliliters of the

10\%

solution,

45

milliliters of the

20\%

solution, and

40

milliliters of the

80\%

solution are needed. The difference between the volume of the

80\%

solution and the

10\%

solution is

40 - 15 = 25

milliliters.

Step-by-Step Solution

1

Define variables for the volume of each acid solution.

Let

x

be the volume of the

10\%

solution,

y

be the volume of the

20\%

solution, and

z

be the volume of the

80\%

solution.

This establishes algebraic representations for the unknowns.

2

Set up a system of linear equations based on the relationships given in the problem statement.

x + y + z = 100

(total volume)

y = 3x

(relationship between the

20\%

and

10\%

solutions)

0.10x + 0.20y + 0.80z = 42.5

(total acid content)

Translating word problems to mathematical equations allows us to solve for the variables systematically.

3

Reduce the system to a single equation in terms of

x

by substituting

y = 3x

and expressing

z

in terms of

x

.

4x + z = 100 \implies z = 100 - 4x

Substitute both into the acid equation:

0.10x + 0.20(3x) + 0.80(100 - 4x) = 42.5

Substitution simplifies the system of equations to a single linear equation with one variable.

4

Solve the simplified linear equation for

x

.

0.70x + 80 - 3.20x = 42.5 \implies -2.50x = -37.5 \implies x = 15

This determines the volume of the

10\%

acid solution.

5

Calculate the volume of the

80\%

solution,

z

.

z = 100 - 4(15) = 40

This determines the volume of the

80\%

acid solution.

6

Find the difference between

z

and

x

.

z - x = 40 - 15 = 25

The question asks for the difference between these two volumes.

Key Concept

Translating and solving systems of linear equations from verbal descriptions (mixture problems).

Estimated Time:2m 30s

Question 1894Question →

Which of the following is a factor of the polynomial $4x^2 - 12xy + 9y^2 - 25$ ?

2x - 3y - 5

2x + 3y - 5

2x - 3y - 25

4x - 6y - 5

2x - 3y + 25

Show answer & explanation

Answer: $2x - 3y - 5$

Answer

The expression

2x - 3y - 5

is a factor of the polynomial.

The polynomial can be factored by first grouping the first three terms as a perfect square trinomial:

4x^2 - 12xy + 9y^2 = (2x - 3y)^2

. This simplifies the expression to

(2x - 3y)^2 - 25

. Since

25 = 5^2

, this is a difference of squares of the form

A^2 - B^2

, which factors into

(A - B)(A + B)

. Substituting

A = 2x - 3y

and

B = 5

gives the factored form

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

. Thus, the option representing

2x - 3y - 5

is a factor.

Step-by-Step Solution

1

Group the first three terms of the polynomial and recognize the perfect square trinomial pattern.

4x^2 - 12xy + 9y^2 = (2x - 3y)^2

The term

4x^2

is

(2x)^2

,

9y^2

is

(3y)^2

, and the middle term

-12xy

is

-2(2x)(3y)

, which matches the perfect square trinomial identity

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

.

2

Rewrite the original expression using the factored trinomial and write 25 as a perfect square.

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

Substituting the factored trinomial and expressing 25 as

5^2

sets up the expression as a difference of squares in the form

A^2 - B^2

.

3

Apply the difference of squares factoring formula

A^2 - B^2 = (A - B)(A + B)

.

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

Substituting

A = 2x - 3y

and

B = 5

into the formula yields the completely factored polynomial.

Key Concept

Factoring by grouping using perfect square trinomials and difference of squares

Estimated Time:1m 0s

Question 1895Question →

For all real numbers $u$ and $v$ , which of the following is equivalent to the expression $\frac{1}{2}(2u - 3v)^2 - \frac{2}{3}u(3u - 9v) - \frac{2}{3}v^2$ ?

\frac{23}{6}v^2

6uv + \frac{23}{6}v^2

\frac{7}{5}v^2

-12uv + \frac{23}{6}v^2

2u^2 - 2u + \frac{23}{6}v^2

Show answer & explanation

Answer: $\frac{23}{6}v^2$

Answer

The simplified equivalent expression is

\frac{23}{6}v^2

.

The correct answer is obtained by expanding

(2u - 3v)^2

to

4u^2 - 12uv + 9v^2

, multiplying it by

\frac{1}{2}

to get

2u^2 - 6uv + \frac{9}{2}v^2

, distributing

-\frac{2}{3}u

to get

-2u^2 + 6uv

, and combining the terms:

(2u^2 - 2u^2) + (-6uv + 6uv) + (\frac{9}{2} - \frac{2}{3})v^2 = \frac{23}{6}v^2

.

Step-by-Step Solution

1

Expand the squared binomial

(2u - 3v)^2

.

4u^2 - 12uv + 9v^2

Using the binomial square formula

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

allows us to expand the expression before applying the outer coefficient.

2

Multiply the expanded binomial by the coefficient

\frac{1}{2}

.

2u^2 - 6uv + \frac{9}{2}v^2

Distributing the constant factor of

\frac{1}{2}

to each term of the expanded binomial.

3

Distribute the term

-\frac{2}{3}u

across the parenthetical expression

(3u - 9v)

.

-2u^2 + 6uv

Multiplying each term inside the parentheses by

-\frac{2}{3}u

, making sure to distribute the negative sign properly:

-\frac{2}{3}u \cdot 3u = -2u^2

and

-\frac{2}{3}u \cdot (-9v) = 6uv

.

4

Combine all terms and group like terms.

2u^2 - 6uv + \frac{9}{2}v^2 - 2u^2 + 6uv - \frac{2}{3}v^2

Write the full expression with all distributed terms to identify and combine like terms.

5

Combine the

u^2

,

uv

, and

v^2

terms.

\frac{23}{6}v^2

Combining the coefficients:

2u^2 - 2u^2 = 0

,

-6uv + 6uv = 0

, and

\frac{9}{2}v^2 - \frac{2}{3}v^2 = (\frac{27}{6} - \frac{4}{6})v^2 = \frac{23}{6}v^2

.

Key Concept

Simplifying expressions by expanding binomials, distributing negative signs, and combining like terms with fractional coefficients.

Alternative Method

Instead of algebraic expansion, you can substitute simple non-zero values for

u

and

v

(e.g.,

u = 3

and

v = 2

) into the original expression and evaluate it. Then, substitute the same values into the answer choices to find which one yields the same result.

Estimated Time:1m 30s

Question 1896Question →

For what values of the real number $p$ is the inequality $-2|p - 4| \geq -10$ true?

p \leq -1

or

p \geq 9

-9 \leq p \leq 1

-1 \leq p \leq 9

p \leq 9

p \geq 9

Show answer & explanation

Answer: $-1 \leq p \leq 9$

Answer

-1 \leq p \leq 9

The correct answer is the inequality showing that

p

is between

-1

and

9

, inclusive. First, divide both sides of the inequality

-2|p - 4| \geq -10

by

-2

and reverse the inequality sign to obtain

|p - 4| \leq 5

. Next, set up the compound inequality

-5 \leq p - 4 \leq 5

. Finally, add

4

to all parts of the inequality to isolate

p

, which yields

-1 \leq p \leq 9

.

Step-by-Step Solution

1

Divide both sides of the inequality

-2|p - 4| \geq -10

by

-2

.

|p - 4| \leq 5

Dividing by a negative number reverses the direction of the inequality sign.

2

Rewrite the absolute value inequality

|p - 4| \leq 5

as a compound inequality.

-5 \leq p - 4 \leq 5

An inequality of the form

|x| \leq a

for

a \geq 0

is equivalent to

-a \leq x \leq a

.

3

Add

4

to all parts of the compound inequality to isolate

p

.

-1 \leq p \leq 9

Adding a constant to all parts of an inequality preserves the inequality relationships and isolates the variable.

Key Concept

Solving absolute value inequalities involving multiplication or division by a negative number.

Question 1897Question →

A newly designed temperature scale, Scale X, is related to the Celsius scale ( $^{\circ}\text{C}$ ) by a linear equation. Water freezes at $0^{\circ}\text{C}$ , which corresponds to $-15^{\circ}\text{X}$ , and water boils at $100^{\circ}\text{C}$ , which corresponds to $135^{\circ}\text{X}$ . If a chemical reaction must be maintained at a temperature where the reading on Scale X is exactly $2.5$ times the reading on the Celsius scale, what is this temperature in degrees Celsius?

-11.5

-8.2

-15.0

13.6

15.0

Show answer & explanation

Answer: $-15.0$

Answer

-15.0

degrees Celsius

By writing the linear relationship between Scale X (

X

) and Celsius (

C

) as

X = mC + k

, we determine the constants using the given coordinates:

(0, -15)

yields

k = -15

, and

(100, 135)

yields

m = 1.5

. The resulting equation is

X = 1.5C - 15

. We then substitute the given condition

X = 2.5C

, resulting in

2.5C = 1.5C - 15

. Subtracting

1.5C

from both sides gives

C = -15

.

Step-by-Step Solution

1

Set up the general linear equation relating Scale X (

X

) and Celsius (

C

).

X = mC + k

Since the relationship is linear, it can be represented by a slope-intercept linear model.

2

Use the freezing point of water to find the y-intercept

k

.

When

C = 0

,

X = -15

, so

-15 = m(0) + k \implies k = -15

.

The freezing point of water provides the point

(0, -15)

on the linear graph.

3

Use the boiling point of water to find the slope

m

.

When

C = 100

,

X = 135

, so

135 = m(100) - 15 \implies 150 = 100m \implies m = 1.5

.

The boiling point of water provides the second point

(100, 135)

to determine the rate of change.

4

Substitute the condition

X = 2.5C

into the linear equation and solve for

C

.

2.5C = 1.5C - 15 \implies 1.0C = -15 \implies C = -15

.

This isolates the Celsius variable to find the temperature where the Scale X value is exactly

2.5

times the Celsius value.

Key Concept

Formulating and solving a linear equation from word-problem constraints and coordinate pairs.

Alternative Method

Instead of deriving the full equation, you can test the options directly. For example, check

-15^{\circ}\text{C}

. The distance from freezing (

0^{\circ}\text{C}

) to

-15^{\circ}\text{C}

is

-15

units. Since Scale X changes by

1.5

units for every

1

unit of Celsius (calculated from a change of

150

on Scale X for

100

on Celsius), Scale X will change by

1.5 \times (-15) = -22.5

units from its freezing point value of

-15

. This yields

-15 - 22.5 = -37.5^{\circ}\text{X}

. Checking the ratio:

\frac{-37.5}{-15} = 2.5

, which matches the given condition.

Estimated Time:2m 0s

Question 1898Question →

The rational expression $\frac{x^2 - 4}{2x^2 + 5x - 3}$ is undefined for two real values of $x$ . If the smaller value is $a$ and the larger value is $b$ , what are the values of $a$ and $b$ ?

Fill in the blanks below

a =

b =

Show answer & explanation

Answer

The smaller value is

-3

and the larger value is

1/2

(or

0.5

).

A rational expression is undefined when the denominator is equal to zero. To find these values, set the denominator

2x^2 + 5x - 3

equal to

0

and solve for

x

. Factoring the quadratic yields

(2x - 1)(x + 3) = 0

. Setting each factor to zero gives

x = 1/2

and

x = -3

. Since

-3

is less than

1/2

, the smaller value is

-3

and the larger value is

1/2

(or

0.5

).

Step-by-Step Solution

1

Identify the condition that makes a rational expression undefined.

The rational expression is undefined when its denominator is equal to zero:

2x^2 + 5x - 3 = 0

.

Division by zero is undefined in the set of real numbers.

2

Factor the quadratic expression in the denominator.

2x^2 + 5x - 3 = (2x - 1)(x + 3) = 0

.

Factoring allows us to find the roots of the quadratic equation using the zero product property.

3

Solve for the roots of the factored equation.

2x - 1 = 0 \Rightarrow x = 1/2

, and

x + 3 = 0 \Rightarrow x = -3

.

Setting each linear factor to zero determines the values of

x

that make the denominator zero.

4

Assign the values to the variables based on the inequality constraint.

Since

-3 < 1/2

, the smaller value is

a = -3

and the larger value is

b = 1/2

(or

0.5

).

The question specifies that

a

is the smaller value and

b

is the larger value.

Key Concept

Identifying domain restrictions of rational expressions by finding where the denominator is equal to zero.

Estimated Time:1m 30s

Question 1899Question →

A craft shop sells handmade candles. The price of a large candle is $3$ dollars more than twice the price of a small candle. If a large candle costs $15$ dollars, what is the price, in dollars, of a small candle?

Show answer & explanation

Answer: 6

Answer

The price of a small candle is

6

dollars.

By letting

s

represent the price of a small candle, the price of a large candle is

2s + 3

. Since the large candle costs

15

dollars, we write the equation

2s + 3 = 15

. Subtracting

3

from both sides gives

2s = 12

, and dividing by

2

yields

s = 6

. Therefore, the price of a small candle is

6

dollars.

Step-by-Step Solution

1

Define the variable and translate the verbal description into an algebraic expression.

Let

s

be the price of a small candle. The price of a large candle is expressed as

2s + 3

.

Translating 'twice the price of a small candle' to

2s

and '3 more than' to

+ 3

allows us to represent the large candle's cost algebraically.

2

Formulate an equation by setting the expression equal to the known cost of the large candle.

2s + 3 = 15

The problem states that the large candle costs

15

dollars.

3

Solve the equation for the variable

s

.

s = 6

Subtracting

3

from both sides gives

2s = 12

. Dividing both sides by

2

isolates

s

, resulting in

6

.

Key Concept

Translating verbal statements into linear equations and solving for a single variable.

Alternative Method

We can solve the problem by working backward from the price of the large candle. Since the large candle (

15

dollars) is

3

dollars more than twice the small candle's price, we subtract

3

dollars to find twice the price of the small candle:

15 - 3 = 12

dollars. Then, since

12

dollars is twice the price of the small candle, we divide by

2

to find the price of a single small candle:

12 \div 2 = 6

dollars.

Estimated Time:45s

Question 1900Question →

A quadratic equation of the form $a x^2 + b x + c = 0$ , where $a$ , $b$ , and $c$ are real constants and $a > 0$ , has a discriminant of $37$ . If the sum of the roots of this equation is $5.5$ and the product of the roots is $5.25$ , what is the value of the coefficient $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of the coefficient

a

is

2

.

By applying Vieta's formulas, we can write

b = -5.5a

and

c = 5.25a

. Plugging these into the discriminant formula gives

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

. Setting the discriminant to

37

results in

9.25a^2 = 37 \implies a^2 = 4

. Since the problem specifies

a > 0

, taking the positive square root gives

a = 2

.

Step-by-Step Solution

1

Express the coefficients

b

and

c

in terms of

a

using Vieta's formulas.

b = -5.5a

and

c = 5.25a

The sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

2

Substitute the expressions for

b

and

c

into the discriminant formula

D = b^2 - 4ac

.

D = 9.25a^2

Substituting the terms yields

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

.

3

Equate the discriminant expression to

37

and solve for

a

.

a = 2

Since

D = 37

, we write

9.25a^2 = 37 \implies a^2 = 4

. Because

a

must be positive, we find

a = 2

.

Key Concept

Using the properties of quadratic roots (Vieta's formulas) and the definition of the discriminant to solve for coefficients.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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