Question

$5x {}^{2} + 6 \sqrt{5 } x + 9 = 0$
by completing the square method only ​

Answer 1

Answer :-

Root of equation is - 3√5 / 5

Explanation :-

$5x^{2} + 6 \sqrt{5} x + 9 = 0 \\ \\ \implies 5x^{2} + 6 \sqrt{5} x = - 9 \\ \\ \tt dividing \ throughout \ by \ 5 \\ \\ \implies \dfrac{5x^{2} }{5} + \dfrac{6 \sqrt{5}x}{5} = - \dfrac{9}{5} \\ \\ \implies x^{2} + 2(x) \bigg( \frac{6 \sqrt{5} }{10} \bigg) = - \dfrac{9}{5} \\ \\ \tt adding \ \bigg(\dfrac{6 \sqrt{5} }{10} \bigg)^{2} on \ both \ sides \\ \\ \ \implies x^{2} + 2(x) \bigg( \dfrac{6 \sqrt{5} }{10} \bigg) + \bigg(\dfrac{6 \sqrt{5} }{10} \bigg)^{2} = - \dfrac{9}{5} + \bigg(\dfrac{6 \sqrt{5} }{10} \bigg)^{2} \\ \\ \implies \bigg(x + \dfrac{6 \sqrt{5} }{10} \bigg)^{2} = - \dfrac{9}{5} + \bigg(\dfrac{6 \sqrt{5} }{10} \bigg)^{2} \\ \\ \sf \{ \because {a}^{2} + 2ab + {b}^{2} = (a + b)^{2} \}$

$\implies \bigg(x + \dfrac{6 \sqrt{5} }{10} \bigg)^{2} = - \dfrac{9}{5} + \dfrac{18}{10} \\ \\ \implies \bigg(x + \dfrac{6 \sqrt{5} }{10} \bigg)^{2} = 0 \\ \\ \implies x + \dfrac{6 \sqrt{5} }{10} = 0 \\ \\ \implies x = - \dfrac{6 \sqrt{5} }{10} \\ \\ \implies x = - \dfrac{3 \sqrt{10} }{5}$

∴ Root of the equation is - 3√10/5