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