Find the roots of 2x^{2}+x-4=0 quadratic equations, if they exist, by the method of completing the square ?

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Solution:

⇒ Given

$Find the roots of 2x^{2}+x-4=0 quadratic equations, if they exist, by the method of completing the square ?,Chapter-4,Exercise-4.3,Math- Class 10,$

∴a = 2, b= 1, c = -4

⇒From Discriminant Formula

$\large =b^2-4ac$

$\large =1^2-4\times 2\times \left ( -4 \right )$

$\large =1+32$

$\large \Rightarrow 33\geq0$

⇒Equation have two real roots.

⇒Finding Real Roots By Completing Square Method

$Find the roots of 2x^{2}+x-4=0 quadratic equations, if they exist, by the method of completing the square ?,Chapter-4,Exercise-4.3,Math- Class 10,$

$\large \large \Rightarrow \left ( \sqrt{2}x+\frac{1}{2\sqrt{2}} \right )^{2}=\left ( \frac{1}{2\sqrt{2}} \right )^{2}+4$

$\large \large \Rightarrow \left ( \sqrt{2}x+\frac{1}{2\sqrt{2}} \right )^{2}=\frac{1}{8}+4$

$\large \large \Rightarrow \left ( \sqrt{2}x+\frac{1}{2\sqrt{2}} \right )^{2}=\frac{1+32}{8}$

$\large \large \Rightarrow \left ( \sqrt{2}x+\frac{1}{2\sqrt{2}} \right )^{2}=\frac{33}{8}$

$\large \large \Rightarrow \left ( \sqrt{2}x+\frac{1}{2\sqrt{2}} \right )^{2}=\left ( \frac{\sqrt{33}}{2\sqrt{2}}\right )^{2}$

$\large \large \Rightarrow \sqrt{2}x+\frac{1}{2\sqrt{2}}=\pm \frac{\sqrt{33}}{2\sqrt{2}}$

$\large \large \Rightarrow \sqrt{2}x=\pm \frac{\sqrt{33}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}$

$\large \large \Rightarrow \sqrt{2}x=\frac{\sqrt{33}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}},-\frac{\sqrt{33}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}$

$\large \large \Rightarrow \sqrt{2}x=\frac{\sqrt{33}-1}{2\sqrt{2}},\frac{-\sqrt{33}-1}{2\sqrt{2}}$

$\large \large \Rightarrow x=\frac{\sqrt{33}-1}{\sqrt{2}\times 2\sqrt{2}},\frac{-\sqrt{33}-1}{\sqrt{2}\times 2\sqrt{2}}$

$\large \large \Rightarrow x=\frac{\sqrt{33}-1}{4},\frac{-\sqrt{33}-1}{4}$