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Mathematics

Solve the following equation by factorisation:

3x−8x=23x - \dfrac{8}{x} = 2

Quadratic Equations

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Answer

Given,

3x−8x=2⇒3x2−8x=2⇒3x2−8=2x⇒3x2−2x−8=0 (Writing as ax2+bx+c=0)⇒3x2−6x+4x−8=0⇒3x(x−2)+4(x−2)=0⇒(3x+4)(x−2)=0 (Factorising left side) ⇒3x+4=0 or x−2=0 (Zero-product rule) ⇒3x=−4 or x=2x=−43 or x=23x - \dfrac{8}{x} = 2 \\[1em] \Rightarrow \dfrac{3x^2 - 8}{x} = 2 \\[1em] \Rightarrow 3x^2 - 8 = 2x \\[1em] \Rightarrow 3x^2 - 2x - 8 = 0 \text{ (Writing as } ax^2 + bx + c = 0) \\[1em] \Rightarrow 3x^2 - 6x + 4x - 8 = 0 \\[1em] \Rightarrow 3x(x - 2) + 4(x - 2) = 0 \\[1em] \Rightarrow (3x + 4)(x - 2) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow 3x + 4 = 0 \text{ or } x - 2 = 0 \text{ (Zero-product rule) } \\[1em] \Rightarrow 3x = -4 \text { or } x = 2 \\[1em] x = -\dfrac{4}{3} \text{ or } x = 2 \\[1em]

Hence, the roots of given equation are −43-\dfrac{4}{3} , 2.

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