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Mathematics

Solve the following equation by factorisation:

a2x2 + 2ax + 1 = 0 , a ≠ 0.

Quadratic Equations

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Answer

Given,

a2x2+2ax+1=0a2x2+ax+ax+1=0ax(ax+1)+1(ax+1)=0(ax+1)(ax+1)=0 (Factorising left side) ax+1=0 (Zero-product rule) ax=1x=1aa^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 1a,1a-\dfrac{1}{a} , -\dfrac{1}{a}

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