Question:
Solve the following quadratic equations by factorization:
$x^{2}+\left(a+\frac{1}{a}\right) x+1=0$
Solution:
We have been given
$x^{2}+\left(a+\frac{1}{a}\right) x+1=0$
Therefore,
$x^{2}+a x+\frac{1}{a} x+1=0$
$x(x+a)+\frac{1}{a}(x+a)=0$
$\left(x+\frac{1}{a}\right)(x+a)=0$
Therefore,
$x+\frac{1}{a}=0$
$x=-\frac{1}{a}$
or,
$x+a=0$
$x=-a$
Hence, $x=-\frac{1}{a}$ or $x=-a$.
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