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Mathematics

Solve the following equation by factorisation:

a2x2 + (a2 + b2)x + b2 = 0, a ≠ 0.

Quadratic Equations

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Answer

Given,

a2x2+(a2+b2)x+b2=0a2x2+a2x+b2x+b2=0a2x(x+1)+b2(x+1)=0(a2x+b2)(x+1)=0 (Factorising left side) a2x+b2=0 or x+1=0 (Zero-product rule) a2x=b2 or x=1x=b2a2 or x=1a^2x^2 + (a^2 + b^2)x + b^2 = 0 \\[0.5em] \Rightarrow a^2x^2 + a^2x + b^2x + b^2 = 0 \\[0.5em] \Rightarrow a^2x(x + 1) + b^2(x + 1) = 0 \\[0.5em] \Rightarrow (a^2x + b^2)(x + 1) = 0 \text{ (Factorising left side) } \\[0.5em] \Rightarrow a^2x + b^2 = 0 \text{ or } x + 1 = 0 \text{ (Zero-product rule) } \\[0.5em] \Rightarrow a^2x = -b^2 \text{ or } x = -1 \\[0.5em] \Rightarrow x = -\dfrac{b^2}{a^2} \text{ or } x =-1 \\[0.5em]

Hence, the roots of given equation are b2a2,1.-\dfrac{b^2}{a^2}, -1.

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