Factor \({x}^{2}+8x+7\).
Ask: Which two numbers add up to \(8\) and multiply to \(7\)?
Rewrite the expression using the above.
\[(x+1)(x+7)\]
\[\frac{x-2}{(x+1)(x+7)}=\frac{2x-5}{{x}^{2}-49}-\frac{x-2}{x_2-6x-7}\]
Rewrite \({x}^{2}-49\) in the form \({a}^{2}-{b}^{2}\), where \(a=x\) and \(b=7\).
\[\frac{x-2}{(x+1)(x+7)}=\frac{2x-5}{{x}^{2}-{7}^{2}}-\frac{x-2}{x_2-6x-7}\]
Use Difference of Squares : \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{x-2}{(x+1)(x+7)}=\frac{2x-5}{(x+7)(x-7)}-\frac{x-2}{x_2-6x-7}\]
Multiply both sides by the Least Common Denominator: \((x+1)(x+7)(x-7)(x_2-6x-7)\).
\[(x-2)(x-7)(x_2-6x-7)=(2x-5)(x+1)(x_2-6x-7)-(x-2)(x+1)(x+7)(x-7)\]
Simplify.
\[x_2{x}^{2}-6{x}^{3}+47{x}^{2}-9x_2x-21x+14x_2-98=2x_2{x}^{2}-11{x}^{3}+55{x}^{2}-3x_2x+2x-5x_2-63-{x}^{4}\]
Move all terms to one side.
\[x_2{x}^{2}-6{x}^{3}+47{x}^{2}-9x_2x-21x+14x_2-98-2x_2{x}^{2}+11{x}^{3}-55{x}^{2}+3x_2x-2x+5x_2+63+{x}^{4}=0\]
Simplify \(x_2{x}^{2}-6{x}^{3}+47{x}^{2}-9x_2x-21x+14x_2-98-2x_2{x}^{2}+11{x}^{3}-55{x}^{2}+3x_2x-2x+5x_2+63+{x}^{4}\) to \(-x_2{x}^{2}+5{x}^{3}-8{x}^{2}-6x_2x-23x+19x_2-35+{x}^{4}\).
\[-x_2{x}^{2}+5{x}^{3}-8{x}^{2}-6x_2x-23x+19x_2-35+{x}^{4}=0\]
No root was found algebraically. However, the following root(s) were found by numerical methods.
\[x=-5.868540,2.731069\]
x=-5.8685401916504,2.7310691833496
