Factor \({x}^{2}+5x-14\).
Ask: Which two numbers add up to \(5\) and multiply to \(-14\)?
Rewrite the expression using the above.
\[(x-2)(x+7)\]
\[\frac{(x-2)(x+7)}{4{x}^{2}}\times \frac{8x}{{x}^{2}-49}\]
Rewrite \({x}^{2}-49\) in the form \({a}^{2}-{b}^{2}\), where \(a=x\) and \(b=7\).
\[\frac{(x-2)(x+7)}{4{x}^{2}}\times \frac{8x}{{x}^{2}-{7}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(x-2)(x+7)}{4{x}^{2}}\times \frac{8x}{(x+7)(x-7)}\]
Cancel \(x+7\).
\[\frac{x-2}{4{x}^{2}}\times \frac{8x}{x-7}\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{(x-2)\times 8x}{4{x}^{2}(x-7)}\]
Regroup terms.
\[\frac{8x(x-2)}{4{x}^{2}(x-7)}\]
Take out the constants.
\[\frac{8}{4}\times \frac{x(x-2)}{{x}^{2}(x-7)}\]
Simplify \(\frac{8}{4}\) to \(2\).
\[2\times \frac{x(x-2)}{{x}^{2}(x-7)}\]
Simplify.
\[\frac{2(x-2)}{x(x-7)}\]
(2*(x-2))/(x*(x-7))
