Solve the following equations by using the method of completing t… - Vidyadip

Question

Solve the following equations by using the method of completing the square:
$\frac{2}{\text{x}^2}-\frac{5}{\text{x}}+2=0$

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Answer

$\frac{2}{\text{x}^2}-\frac{5}{\text{x}}+2=0$
$\Rightarrow\frac{2-\text{5x}+\text{2x}^2}{\text{x}^2}=0$
$\Rightarrow 2x^2 - 5x + 2 = 0$
$\Rightarrow 4x^2 - 10x + 4 = 0$ (Multiplying both sides by 2)
$\Rightarrow 4x^2 - 10x = -4$
$\Rightarrow(\text{2x})^2-2\times\text{2x}\times\frac{5}{2}+\Big(\frac{5}{2}\Big)^2\\=-4+\Big(\frac{5}{2}\Big)^2$ $\Big[$Adding $\Big(\frac{5}2{}\Big)^2$ on both sides$\Big]$
$\Rightarrow\Big(\text{2x}-\frac{5}{2}\Big)^2$
$=-4+\frac{25}{4}$
$=\frac{9}{4}=\Big(\frac{3}{2}\Big)^2$
$\Rightarrow\text{2x}-\frac{5}{2}=\pm\frac{3}{2}$ (Taking square root on both sides)
$\Rightarrow\text{2x}-\frac{5}{2}=\frac{3}{2}$ or $\text{2x}-\frac{5}{2}=-\frac{3}{2}$
$\Rightarrow\text{2x}=\frac{3}{2}+\frac{5}{2}=\frac{8}{2}=4$ or $\text{2x}=-\frac{3}{2}+\frac{5}{2}=\frac{2}{2}=1$
$\Rightarrow x = 2$ or $\text{x}=\frac{1}{2}$
Hence, 2 and $\frac{1}{2}$ are the roots of the given equation.

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