Algebra II : Understanding Quadratic Equations

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Binomials

Expand:

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

Possible Answers:

\(\displaystyle 3x^{2}-29x-18\)

\(\displaystyle 3x^{2}-18x-29\)

\(\displaystyle 3x^{2}-25x-18\)

\(\displaystyle 3x^{2}-18x-25\)

None of the other answers

Correct answer:

\(\displaystyle 3x^{2}-25x-18\)

Explanation:

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

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

or \(\displaystyle 3x^{2}-25x-18\).

Example Question #11 : Trinomials

Evaluate the following:

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

Possible Answers:

\(\displaystyle 2x^2+3x+8\)

\(\displaystyle 2x^2-3x-8\)

\(\displaystyle x^2+3x-8\)

\(\displaystyle 2x^2-3x\)

Correct answer:

\(\displaystyle 2x^2-3x-8\)

Explanation:

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

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

\(\displaystyle \frac{4}{2}x^2 -3x - 8\)

\(\displaystyle 2x^2 -3x - 8\)

 

Example Question #12 : Understanding Quadratic Equations

Simplify.

\(\displaystyle \small \small \frac{x^2+5x+6}{x^3+2x^2-16x-32}\)

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

Example Question #13 : Understanding Quadratic Equations

Simplfy.

\(\displaystyle \small \frac{x^2-2x-15}{x^2-3x-10}\)

Possible Answers:

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

\(\displaystyle \small \frac{x+3}{x+2}\)

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

\(\displaystyle \small \small \frac{x+2}{x+3}\)

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

Correct answer:

\(\displaystyle \small \frac{x+3}{x+2}\)

Explanation:

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

Example Question #14 : Understanding Quadratic Equations

Expand.

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

Possible Answers:

\(\displaystyle \small x^2+4x+8\)

\(\displaystyle \small \small 2x^2+12x-16\)

\(\displaystyle \small 2x^2+12x+16\)

\(\displaystyle \small \small x^2+4x-8\)

\(\displaystyle \small 2x+12x+16\)

Correct answer:

\(\displaystyle \small 2x^2+12x+16\)

Explanation:

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

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

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

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

Example Question #15 : Understanding Quadratic Equations

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

Possible Answers:

\(\displaystyle -24\)

\(\displaystyle -15\)

\(\displaystyle 15\)

\(\displaystyle -1\)

\(\displaystyle 18\)

Correct answer:

\(\displaystyle -1\)

Explanation:

Use the FOIL method to simplify the binomial.

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

\(\displaystyle (ax)(2ax)+(ax)(3b)+(b)(2ax)+(b)(3b)\)

Simplify the terms.

\(\displaystyle 2a^2x^2+5abx+3b^2=8x^2+30x+27\)

Notice that the coefficients can be aligned to the unknown variables.  Solve for \(\displaystyle a\) and \(\displaystyle b\).

\(\displaystyle 2a^2 = 8 \rightarrow a^2 = 4 \rightarrow a=2\)

\(\displaystyle 3b^2 = 27\rightarrow b^2=9 \rightarrow b=3\)

\(\displaystyle a-b = 2-3 = -1\)

The answer is:  \(\displaystyle -1\)

Example Question #16 : Understanding Quadratic Equations

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

Possible Answers:

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

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

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

\(\displaystyle \textup{The answer is not given.}\)

\(\displaystyle 3x^4-3x^3-x^2-2x-3\)

Correct answer:

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

Explanation:

Multiply each term of the first trinomial by second trinomial.

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

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

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

Add and combine like-terms.

The answer is:  \(\displaystyle 3x^4-4x^3+x^2-2x-3\)

Example Question #17 : Understanding Quadratic Equations

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

Possible Answers:

\(\displaystyle \frac{3}{16}x-\frac{3}{16}\)

\(\displaystyle \frac{1}{8}x+\frac{5}{8}\)

\(\displaystyle \frac{1}{16}x-\frac{5}{16}\)

\(\displaystyle \textup{The expression cannot be factored.}\)

\(\displaystyle \frac{1}{16}x+\frac{5}{16}\)

Correct answer:

\(\displaystyle \frac{1}{8}x+\frac{5}{8}\)

Explanation:

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

\(\displaystyle \frac{8x-10+2x^2}{-16+16x} =\frac{2x^2+8x-10}{16x-16}\)

Factor a 2 in the numerator.

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

Factor the term in parentheses.

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

Factor the denominator.

\(\displaystyle 16x-16 = 16(x-1)\)

Divide the numerator with the denominator.

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

The expression becomes:

 \(\displaystyle \frac{x+5}{8}\)

The answer is:  \(\displaystyle \frac{1}{8}x+\frac{5}{8}\)

Example Question #18 : Understanding Quadratic Equations

Solve for x:

 

\(\displaystyle \frac{6}{x}=\frac{x+2}{4}\)

Possible Answers:


None of the other answers

\(\displaystyle -6,4\)

\(\displaystyle 12,-2\)

\(\displaystyle -12,2\)

\(\displaystyle -4,6\)

Correct answer:

\(\displaystyle -6,4\)

Explanation:

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

\(\displaystyle 24=x^2+2x\)

 

After subtracting \(\displaystyle 24\) from each side the equation looks like:

 

\(\displaystyle 0=x^2+2x-24\)

 

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

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

 

Both \(\displaystyle 4\) and \(\displaystyle 6\) satisfy the above equation and are therefore the correct answers. 

Example Question #1 : Simplifying Polynomials

Expand this expression:

\(\displaystyle (x-4)\)\(\displaystyle (2x+4)\)

Possible Answers:

\(\displaystyle 2x^2 + 4x + 16\)

\(\displaystyle 2x^2 - 4x + 16\)

\(\displaystyle 2x^2 - 4x - 16\)

\(\displaystyle 2x^2 + 4x - 16\)

\(\displaystyle 2x - 4x - 16\)

Correct answer:

\(\displaystyle 2x^2 - 4x - 16\)

Explanation:

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

\(\displaystyle x * 2x = 2x^2\)

\(\displaystyle x * 4 = 4x\)

\(\displaystyle -4 * 2x = -8x\)

\(\displaystyle -4 * 4 = -16\)

Put all of these terms together:

\(\displaystyle 2x^2 + 4x - 8x - 16\)

Combine like terms:

\(\displaystyle 2x^2 - 4x - 16\)

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