Question
Solve the following quadratic by factorization:$a(x^2 + 1) - x(a^2 + 1) = 0$
✓
Answer
We have
$a\left(x^2+1\right)-x\left(a^2+1\right)=0$
$\Rightarrow a\left(x^2-1\right)-a^2 x-x+a=0$
${\left[\because a \times a=a^2 \Rightarrow a^2=-a^2 \times-1-\left(a^2+1\right)=a^2-1\right]}$
$\Rightarrow a \times(x-a)-1(x-a)=0$
$\Rightarrow(x-a)(a x-1)=0$
$\Rightarrow x-a=0 \text { or } a x-1=0$
$\Rightarrow x=a \text { or } x=\frac{1}{a}$
$\therefore x=a$ and $x=\frac{1}{a}$ are the two roots of the given equation.
$a\left(x^2+1\right)-x\left(a^2+1\right)=0$
$\Rightarrow a\left(x^2-1\right)-a^2 x-x+a=0$
${\left[\because a \times a=a^2 \Rightarrow a^2=-a^2 \times-1-\left(a^2+1\right)=a^2-1\right]}$
$\Rightarrow a \times(x-a)-1(x-a)=0$
$\Rightarrow(x-a)(a x-1)=0$
$\Rightarrow x-a=0 \text { or } a x-1=0$
$\Rightarrow x=a \text { or } x=\frac{1}{a}$
$\therefore x=a$ and $x=\frac{1}{a}$ are the two roots of the given equation.
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