Find the roots of the equation $a^2b^2x^2 + b^2x - a^2x - 1 = 0$.
Step-by-Step Solution
Key Concept: Group terms by factorisation: $b^2x(a^2x + 1) - 1(a^2x + 1) = 0$.
Stepwise Solution:
$a^2b^2x^2 + b^2x - a^2x - 1 = 0 \Rightarrow b^2x(a^2x + 1) - 1(a^2x + 1) = 0$. [1.0 Mark]
$(b^2x - 1)(a^2x + 1) = 0$. [1.0 Mark]
Either $b^2x - 1 = 0 \Rightarrow x = \dfrac{1}{b^2}$ or $a^2x + 1 = 0 \Rightarrow x = -\dfrac{1}{a^2}$.
Roots are $\dfrac{1}{b^2}$ and $-\dfrac{1}{a^2}$. [1.0 Mark]
Marking Scheme:
• Grouping terms: 1.0 Mark
• Factorising to $(b^2x - 1)(a^2x + 1) = 0$: 1.0 Mark
• Finding roots $x = 1/b^2$ and $x = -1/a^2$: 1.0 Mark
Correct Answer: