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Algebra II : Understanding Quadratic Equations

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Example Questions

Example Question #1 : Binomials

Expand:

(x-9)(3x+2)

Possible Answers:

$3x^{2}-18x-25$

None of the other answers

$3x^{2}-18x-29$

$3x^{2}-29x-18$

$3x^{2}-25x-18$

Correct answer:

$3x^{2}-25x-18$

Explanation:

To multiple these binomials, you can use the FOIL method to multiply each of the expressions individually.This will give you

(x)(3x)+(x)(2)+(-9)(3x)+(-9)(2)

or $3x^{2}-25x-18$ .

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Example Question #2 : How To Add Trinomials

Evaluate the following:

$\left(\frac{1}{2}x^2 +2x - 4\right) + \left(\frac{3}{2}x^2 -5x -4\right)$

Possible Answers:

2x^2+3x+8

2x^2-3x-8

2x^2-3x

x^2+3x-8

Correct answer:

2x^2-3x-8

Explanation:

To add these two trinomials, you will first begin by combining like terms. You have two terms with x^2 , two terms with , and two terms with no variable. For the two fractions with x^2 , you can immediately add because they have common denominators:

$(\frac{1}{2}x^2 +2x - 4) + (\frac{3}{2}x^2 -5x -4)$

$\frac{4}{2}x^2 -3x - 8$

2x^2 -3x - 8

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Example Question #11 : Quadratic Equations And Inequalities

Simplify.

$\small \small \frac{x^2+5x+6}{x^3+2x^2-16x-32}$

Possible Answers:

$\small \small \frac{x-2}{(x-4)(x+4)}$

$\small \small \frac{x+3}{(x+4)(x+4)}$

$\small \small \frac{x-3}{(x-4)(x+4)}$

$\small \small \frac{x+3}{(x-2)(x+4)}$

$\small \frac{x+3}{(x-4)(x+4)}$

Correct answer:

$\small \frac{x+3}{(x-4)(x+4)}$

Explanation:

Factoring the expression gives $\small \frac{}{}$ $\small \small \frac{(x+3)(x+2)}{(x-4)(x+4)(x+2)}$ . Values that are in both the numerator and denominator can be cancelled. By cancelling $\small x+2$ , the expression becomes $\small \small \frac{(x+3)}{(x-4)(x+4)}$ .

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Example Question #12 : Quadratic Equations And Inequalities

Simplfy.

$\small \frac{x^2-2x-15}{x^2-3x-10}$

Possible Answers:

$\small \small \frac{x+2}{x+3}$

$\small \small \frac{x-3}{x+2}$

$\small \small \frac{x+3}{x-2}$

$\small \frac{x+3}{x+2}$

$\small \small \small \frac{x-2}{x+3}$

Correct answer:

$\small \frac{x+3}{x+2}$

Explanation:

By factoring the equation you get $\small \small \small \frac{(x+3)(x-5)}{(x+2)(x-5)}$ . Values that are in both the numerator and denominator can be cancelled. Cancelling the $\small x-5$ values gives $\small \small \small \frac{x+3}{x+2}$ .

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Example Question #13 : Quadratic Equations And Inequalities

Expand.

$\small 2(x+2)(x+4)$

Possible Answers:

$\small x^2+4x+8$

$\small \small 2x^2+12x-16$

$\small 2x+12x+16$

$\small 2x^2+12x+16$

$\small \small x^2+4x-8$

Correct answer:

$\small 2x^2+12x+16$

Explanation:

By foiling the binomials, multiplying the firsts, then the outers, followed by the inners and lastly the lasts, the expression you get is:

$\small 2(x^2+4x+2x+8)$ .

However, the expression can not be considered simplified in this state.

Distributing the two and adding like terms gives $\small 2x^2+12x+16$ .

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Example Question #12 : Understanding Quadratic Equations

If (ax+b)(2ax+3b)=8x^2+30x+27 , what is the value of a-b ?

Possible Answers:

Correct answer:

Explanation:

Use the FOIL method to simplify the binomial.

(a+b)(c+d) = ac+ad+bc+bd

(ax)(2ax)+(ax)(3b)+(b)(2ax)+(b)(3b)

Simplify the terms.

2a^2x^2+5abx+3b^2=8x^2+30x+27

Notice that the coefficients can be aligned to the unknown variables. Solve for and .

$2a^2 = 8 \rightarrow a^2 = 4 \rightarrow a=2$

$3b^2 = 27\rightarrow b^2=9 \rightarrow b=3$

a-b = 2-3 = -1

The answer is:

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Example Question #15 : Quadratic Equations And Inequalities

Multiply: (x^2-x-1)(3x^2-x+3)

Possible Answers:

3x^4-4x^3+x^2-2x-3

$\textup{The answer is not given.}$

3x^4-4x^3+4x^2-4x-3

3x^4-3x^3-x^2-2x-3

3x^4-3x^3-x^2+2x-3

Correct answer:

3x^4-4x^3+x^2-2x-3

Explanation:

Multiply each term of the first trinomial by second trinomial.

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

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

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

Add and combine like-terms.

The answer is: 3x^4-4x^3+x^2-2x-3

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Example Question #13 : Understanding Quadratic Equations

Simplify the function, if possible: $\frac{8x-10+2x^2}{-16+16x}$

Possible Answers:

$\frac{3}{16}x-\frac{3}{16}$

$\textup{The expression cannot be factored.}$

$\frac{1}{16}x+\frac{5}{16}$

$\frac{1}{8}x+\frac{5}{8}$

$\frac{1}{16}x-\frac{5}{16}$

Correct answer:

$\frac{1}{8}x+\frac{5}{8}$

Explanation:

The expression will need to be rearranged from highest to lowest powers in order to be simplified.

$\frac{8x-10+2x^2}{-16+16x} =\frac{2x^2+8x-10}{16x-16}$

Factor a 2 in the numerator.

2x^2+8x-10 = 2(x^2+4x-5)

Factor the term in parentheses.

2(x^2+4x-5) = 2(x-1)(x+5)

Factor the denominator.

16x-16 = 16(x-1)

Divide the numerator with the denominator.

$\frac{2(x-1)(x+5)}{16(x-1)}$

The expression becomes:

$\frac{x+5}{8}$

The answer is: $\frac{1}{8}x+\frac{5}{8}$

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Example Question #17 : Quadratic Equations And Inequalities

Solve for x:

$\frac{6}{x}=\frac{x+2}{4}$

Possible Answers:

-12,2

12,-2

-6,4

-4,6

None of the other answers

Correct answer:

-6,4

Explanation:

The correct answer is x=-6 or x=4 . The first step of the problem is to cross multiply. This will give the following equation:

24=x^2+2x

After subtracting from each side the equation looks like:

0=x^2+2x-24

The expression on the right hand side can be factored into:

0=(x-4)(x+6 )

Both and satisfy the above equation and are therefore the correct answers.

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Example Question #3 : Simplifying Polynomials

Expand this expression:

(x-4) (2x+4)

Possible Answers:

2x - 4x - 16

2x^2 + 4x - 16

2x^2 - 4x - 16

2x^2 - 4x + 16

2x^2 + 4x + 16

Correct answer:

2x^2 - 4x - 16

Explanation:

Use the FOIL method (First, Outer, Inner, Last):

x * 2x = 2x^2

x * 4 = 4x

-4 * 2x = -8x

-4 * 4 = -16

Put all of these terms together:

2x^2 + 4x - 8x - 16

Combine like terms:

2x^2 - 4x - 16

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Algebra II : Understanding Quadratic Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Binomials

Example Question #2 : How To Add Trinomials

Example Question #11 : Quadratic Equations And Inequalities

Example Question #12 : Quadratic Equations And Inequalities

Example Question #13 : Quadratic Equations And Inequalities

Example Question #12 : Understanding Quadratic Equations

Example Question #15 : Quadratic Equations And Inequalities

Example Question #13 : Understanding Quadratic Equations

Example Question #17 : Quadratic Equations And Inequalities

Example Question #3 : Simplifying Polynomials

All Algebra II Resources

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