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