Question

Find the roots of the following quadratic equations:-
(i) ${2x}^2+5\sqrt{3}x+6=0$
(ii) ${3x}^2+2\sqrt{5}x-5=0$
(iii) ${25x}^2+20x+7=0$
(iv) $6x^2-x-2=0$

Answer 1

(i)

$2x^2+5\sqrt{3}x+6=0$

$2x^2+4\sqrt{3}x+\sqrt{3}x+6=0$

$2x(x+2\sqrt{3})+\sqrt{3}(x+2\sqrt{3})=0$

$(x+2\sqrt{3})(2x+\sqrt{3})=0$

$x=-2\sqrt{3},-\frac{\sqrt{3}}{2}$

(ii)

$3x^2+2\sqrt{5}x-5=0$

$3x^2+3\sqrt{5}x-\sqrt{5}x5=0$

$3x(x+\sqrt{5})-\sqrt{5}(x+\sqrt{5})=0$

$(x+\sqrt{5})(3x-\sqrt{5})=0$

$x=-\sqrt{5},\frac{\sqrt{5}}{3}$

(iii)

$25x^2+20x+7=0$

$x^2+\frac{4}{5}x+\frac{7}{25}=0$

$x^2+\frac{4}{5}x+\frac{7}{25}+\frac{4}{25}=0+\frac{4}{25}$

$x^2+\frac{4}{5}x+\frac{4}{25}=-\frac{3}{25}$

${(x+\frac{2}{5})}^2=-\frac{3}{25}$

$x+\frac{2}{5}=\sqrt{-\frac{3}{25}}$

$x=-\frac{2}{5}+\sqrt{-\frac{3}{25}}$

$x=-\frac{2}{5}\pm \sqrt{-\frac{3}{25}}$

(iv)

$6x^2-x-2=0$

$6x^2-4x+3x-2=0$

$2x(3x-2)+1(3x-2)=0$

$(3x-2)(2x+1)=0$

$x=\frac{2}{3},-\frac{1}{2}$