Question

Solve the following quadratic equations by factorization:

$\frac{5+x}{5-x}-\frac{5-x}{5+x}=3\frac{3}{4};x\ne 5,-5$

Answer 1

$$\begin{gathered} \frac{5+x}{5-x}-\frac{5-x}{5+x}=3\frac{3}{4} \\ \Rightarrow \frac{{\left(5+x\right)}^2-{\left(5-x\right)}^2}{\left(5+x\right)\left(5-x\right)}=\frac{15}{4} \\ \Rightarrow \frac{25+x^2+10x-25-x^2+10x}{25-x^2}=\frac{15}{4} \\ \Rightarrow \frac{20x}{25-x^2}=\frac{15}{4} \\ \Rightarrow \frac{4x}{25-x^2}=\frac{3}{4} \\ \Rightarrow 16x=75-3x^2 \\ \Rightarrow 3x^2+16x-75=0 \\ \Rightarrow 3x^2+25x-9x-75=0 \\ \Rightarrow x(3x+25)-3(3x+25)=0 \\ \Rightarrow (x-3)(3x+25)=0 \\ \Rightarrow x-3=0\mathrm{or}3x+25=0 \\ \Rightarrow x=3\mathrm{or}x=-\frac{25}{3} \end{gathered}$$

Hence, the factors are 3 and $-\frac{25}{3}$.