Solve the following equation by factorisation:

a²x² + 2ax + 1 = 0 , a ≠ 0.

Given,

$a^2x^2 + 2ax + 1 = 0 \\[0.5em] \Rightarrow a^2x^2 + ax + ax + 1 = 0 \\[0.5em] \Rightarrow ax(ax + 1) + 1(ax + 1) = 0 \\[0.5em] \Rightarrow (ax + 1)(ax + 1) = 0 \text{ (Factorising left side) } \\[0.5em] \Rightarrow ax + 1 = 0 \text{ (Zero-product rule) }\\[0.5em] \Rightarrow ax = -1 \\[0.5em] \Rightarrow x = -\dfrac{1}{a}$

Hence, the roots of given equation are $-\dfrac{1}{a} , -\dfrac{1}{a}$