Question

Find the roots of the following quadratic equations by factorisation:
(i) $x^2-3x-10=0$
(ii) $2x^2+x-6=0$
(iii) $\sqrt{2}{x}^2+7x+5\sqrt{2}=0$
(iv) $2x^2-x+\frac{1}{8}=0$
(v) $100x^2-20x+1=0$

Answer 1

i)

$x^2-3x-10=0$

$\Rightarrow x^2-5x+2x-10=0$

$\Rightarrow x(x-5)+2(x-5)=0$

$\Rightarrow (x+2)(x-5)=0$

$\Rightarrow x=5,-2$

ii)

${2x}^2+x-6=0$

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

$\Rightarrow 2x(x+2)-3(x+2)=0$

$\Rightarrow (x+2)(2x-3)=0$

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

iii)

${\sqrt{2}x}^2+7x+5\sqrt{2}=0$

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

$\Rightarrow \sqrt{2}x(x+\sqrt{2})+5(x+\sqrt{2})=0$

$\Rightarrow (x+\sqrt{2})(\sqrt{2}x+5)=0$

$\Rightarrow x=-\sqrt{2},\frac{-5}{\sqrt{2}}$

iv)

${2x}^2-x+\frac{1}{8}=0$

$\Rightarrow {16x}^2-8x+1=0$

$\Rightarrow {16x}^2-4x-4x+1=0$

$\Rightarrow 16x(x-\frac{1}{4})-4(x-\frac{1}{4})=0$

$\Rightarrow (x-\frac{1}{4})(16x-4)=0$

$\Rightarrow x=\frac{1}{4},\frac{1}{4}$

v)

${100x}^2-20x+1=0$

$\Rightarrow {100x}^2-10x-10x+1=0$

$\Rightarrow 10x(10x-1)-1(10x-1)=0$

$\Rightarrow (10x-1)(10x-1)=0$

$\Rightarrow x=\frac{1}{10},\frac{1}{10}$