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Simplify completely quantity x squared minus 100 over quantity x squared plus 2 x minus 120 divided quantity 6 x plus 60 over quantity x minus 2.


quantity x minus 1 over 6 open parentheses x plus 6 close parentheses


quantity x minus 2 over 6 open parentheses x plus 10 close parentheses


quantity x minus 2 over 6 open parentheses x plus 12 close parentheses


quantity x minus 1 over 3 open parentheses x plus 12 close parentheses

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JcAlmighty

The answer is quantity x minus 2 over 6 open parentheses x plus 12 close parentheses.


quantity x squared minus 100 (x² - 100) over quantity x squared plus 2 x minus 120 (x² + 2x - 120) divided quantity 6 x plus 60 (6x + 60) over quantity x minus 2 (x - 2) can be expressed as
\(\frac{ x^{2} -100}{ x^{2} +2x-120} / \frac{6x+60}{x-2}\)

Since \(\frac{a}{b} / \frac{c}{d}= \frac{a}{b} * \frac{d}{c}= \frac{a*d}{b*c}\), then:
\(\frac{ x^{2} -100}{ x^{2} +2x-120} / \frac{6x+60}{x-2}= \frac{(x^{2} -100)*(x-2)}{(x^{2} +2x-120)*(6x+60)} =\)

Let's simplify some of the factors:
x² - 100 = x² - 10² = (x - 10)(x + 10)              (since a² - b² = (a - b)(a + b))
 6x + 60 = 6 x + 6 10 = 6 (x+10)
 x² + 2x - 120 =  x² + 12x - 10x - 12 10 = (x x - 10 x) + (12 x - 12 * 10) = 
                        = x(x - 10) + 12(x - 10) = (x - 10)(x + 12)

Now substitute this into the expression:
\(\frac{(x^{2} -100)*(x-2)}{(x^{2} +2x-120)*(6x+60)} = \frac{(x-10)(x+10)(x-2)}{(x-10)(x+12)*6(x10)}\)

We can cancel (x-10)(x+10):
\(\frac{(x-10)(x+10)(x-2)}{(x-10)(x+12)*6(x10)} =\frac{(x-2)}{(x+12)*6} =\frac{(x-2)}{6(x+12)}\)

 \(\frac{x-2}{6(x+12)}\) is quantity x minus 2 over 6 open parentheses x plus 12 close parentheses.

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