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