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Question

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


Solution

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) 
$${ 2x }^{ 2 }+x-6=0$$
$$\Rightarrow { 2x }^{ 2 }+4x-3x-6=0$$
$$\Rightarrow 2x(x+2)-3(x+2)=0$$
$$\Rightarrow (x+2)(2x-3)=0$$
$$\Rightarrow x=\dfrac{3}{2},-2$$

iii) 
$${ \sqrt { 2 } x }^{ 2 }+7x+5\sqrt { 2 } =0$$
$$\Rightarrow { \sqrt { 2 } x }^{ 2 }+2x+5x+5\sqrt { 2 } =0$$
$$\Rightarrow \sqrt { 2 } x(x+\sqrt { 2 } )+5(x+\sqrt { 2 } )=0$$
$$\Rightarrow (x+\sqrt { 2 } )(\sqrt { 2 } x+5)=0$$
$$\Rightarrow x=-\sqrt { 2 } ,\dfrac{-5}{\sqrt { 2 }} $$

iv) 
$${ 2x }^{ 2 }-x+\dfrac18=0$$
$$\Rightarrow { 16x }^{ 2 }-8x+1=0$$
$$\Rightarrow { 16x }^{ 2 }-4x-4x+1=0$$
$$\Rightarrow 16x\left(x-\dfrac14\right)-4\left(x-\dfrac14\right)=0$$

$$\Rightarrow \left(x-\dfrac14\right)(16x-4)=0$$

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

v) 
$${ 100x }^{ 2 }-20x+1=0$$
$$\Rightarrow { 100x }^{ 2 }-10x-10x+1=0$$
$$\Rightarrow { 10 }x(10x-1)-1(10x-1)=0$$
$$\Rightarrow (10x-1)(10x-1)=0$$
$$\Rightarrow x=\dfrac{1}{10},\dfrac{1}{10}$$

Mathematics
RS Agarwal
Standard X

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