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Mathematics

Solve the following equation by factorisation:

3x2+10x+73=0\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0

Quadratic Equations

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Answer

Given,

3x2+10x+73=03x2+7x+3x+73=0x(3x+7)+3(3x+7)=0(x+3)(3x+7)=0 (Factorising left side) x+3=0 or 3x+7=0 (Zero- product rule) x=3 or 3x+7=0x=3 or 3x=7x=3 or x=73x=3 or x=733\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0 \\[1em] \Rightarrow \sqrt{3}x^2 + 7x + 3x + 7\sqrt{3} = 0 \\[1em] \Rightarrow x(\sqrt{3}x + 7) + \sqrt{3}(\sqrt{3}x + 7) = 0 \\[1em] \Rightarrow (x + \sqrt{3})(\sqrt{3}x + 7) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow x + \sqrt{3} = 0 \text{ or } \sqrt{3}x + 7 = 0 \text{ (Zero- product rule) } \\[1em] \Rightarrow x = -\sqrt{3} \text{ or } \sqrt{3}x + 7 = 0 \\[1em] \Rightarrow x = -\sqrt{3} \text{ or } \sqrt{3}x = -7 \\[1em] \Rightarrow x = -\sqrt{3} \text{ or } x = -\dfrac{7}{\sqrt{3}} \\[1em] x = -\sqrt{3} \text{ or } x = -\dfrac{7\sqrt{3}}{3}

Hence, the roots of given equation are 3,733-\sqrt{3}, -\dfrac{7\sqrt{3}}{3}

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