Question

Slove The Algebraic Expression. 2a=(2/x-x)² ​

Answer 1

$\textbf{Given:}$

$\textsf{Equation is}\;\mathsf{2a=\left(\dfrac{2}{x}-x\right)^2}$

$\textbf{To find:}$

$\textsf{Roots of the given polyomial equations}$

$\textbf{Solution:}$

$\mathsf{2a=\left(\dfrac{2}{x}-x\right)^2}$

$\mathsf{\left(\dfrac{2-x^2}{x}\right)^2=2a}$

$\textsf{Taking square root on bothsides, we get}$

$\mathsf{\dfrac{2-x^2}{x}\right=\pm\sqrt{2a}}$

$\mathsf{\dfrac{2-x^2}{x}\right=\pm\sqrt{2a}}$

$\underline{\mathsf{case(i):}}$

$\mathsf{\dfrac{2-x^2}{x}\right=\sqrt{2a}}$

$\mathsf{2-x^2=\sqrt{2a}x}$

$\mathsf{x^2+\sqrt{2a}x=2}$

$\mathsf{x^2+\sqrt{2a}x+\dfrac{a}{2}=2+\dfrac{a}{2}}$

$\mathsf{\left(x+\dfrac{\sqrt{a}}{\sqrt{2}}\right)^2=\dfrac{a+4}{2}}$

$\textsf{Taking square root,}$

$\mathsf{x+\dfrac{\sqrt{a}}{\sqrt{2}}=\pm\sqrt{\dfrac{a+4}{2}}}$

$\implies\boxed{\mathsf{x=-\dfrac{\sqrt{a}}{\sqrt{2}}\pm\sqrt{\dfrac{a+4}{2}}}}$

$\underline{\mathsf{case(ii):}}$

$\mathsf{\dfrac{2-x^2}{x}\right=-\sqrt{2a}}$

$\mathsf{2-x^2=-\sqrt{2a}x}$

$\mathsf{x^2-\sqrt{2a}x=2}$

$\mathsf{x^2-\sqrt{2a}x+\dfrac{a}{2}=2+\dfrac{a}{2}}$

$\mathsf{\left(x-\dfrac{\sqrt{a}}{\sqrt{2}}\right)^2=\dfrac{a+4}{2}}$

$\textsf{Taking square root,}$

$\mathsf{x-\dfrac{\sqrt{a}}{\sqrt{2}}=\pm\sqrt{\dfrac{a+4}{2}}}$

$\implies\boxed{\mathsf{x=\dfrac{\sqrt{a}}{\sqrt{2}}\pm\sqrt{\dfrac{a+4}{2}}}}$

$\textbf{Answer:}$

$\mathsf{Roots\;are\;-\dfrac{\sqrt{a}}{\sqrt{2}}\pm\sqrt{\dfrac{a+4}{2}}\;\;\&\;\;\dfrac{\sqrt{a}}{\sqrt{2}}\pm\sqrt{\dfrac{a+4}{2}}}$

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