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Mathematics

Solve the following equation by factorisation:

43x2+5x23=04\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0

Quadratic Equations

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Answer

Given,

43x2+5x23=043x2+8x3x23=04x(3x+2)3(3x+2)=0(3x+2)(4x3)=0 (Factorising left side) 3x+2=0 or 4x3=0 (Zero-product rule) 3x=2 or 4x=3x=23 or x=34x=23×33 or x=34x=233 or x=344\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0 \\[1em] \Rightarrow 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3} = 0 \\[1em] \Rightarrow 4x(\sqrt{3}x + 2) - \sqrt{3}(\sqrt{3}x + 2) = 0 \\[1em] \Rightarrow (\sqrt{3}x + 2)(4x - \sqrt{3}) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow \sqrt{3}x + 2 = 0 \text{ or } 4x - \sqrt{3} = 0 \text{ (Zero-product rule) } \\[1em] \Rightarrow \sqrt{3}x = -2 \text{ or } 4x = \sqrt{3} \\[1em] \Rightarrow x = -\dfrac{2}{\sqrt{3}} \text{ or } x = \dfrac{\sqrt{3}}{4} \\[1em] \Rightarrow x = -\dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} \text{ or } x = \dfrac{\sqrt{3}}{4} \\[1em] x = -\dfrac{2\sqrt{3}}{3} \text{ or } x = \dfrac{\sqrt{3}}{4}

Hence, the roots of given equation are 233-\dfrac{2\sqrt{3}}{3}, 34.\dfrac{\sqrt{3}}{4}.

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