Question

a(x^2 + 1) = (a^2+ 1) x, a + 0 solve by formula method​

Answer 1

$\textbf{Formula used:}$

$\text{The roots of the quadratic equation ax^2+bx+c=0 are}$

$\boxed{\bf\,x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

$\textbf{Given:}\;a(x^2+1)=(a^2+1)x$

$\implies\;ax^2-(a^2+1)x+a=0$

$\textbf{To find: Roots by formula method}$

$x=\displaystyle\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\displaystyle\frac{(a^2+1)\pm\sqrt{(a^2+1)^2-4(a)(a)}}{2a}$

$x=\displaystyle\frac{(a^2+1)\pm\sqrt{a^4+1+2a^2-4a^2}}{2a}$

$x=\displaystyle\frac{(a^2+1)\pm\sqrt{a^4+1-2a^2}}{2a}$

$x=\displaystyle\frac{(a^2+1)\pm\sqrt{(a^2-1)^2}}{2a}$

$x=\displaystyle\frac{(a^2+1)\pm(a^2-1)}{2a}$

$x=\displaystyle\frac{(a^2+1)+(a^2-1)}{2a},\;\frac{(a^2+1)-(a^2-1)}{2a}$

$x=\displaystyle\frac{2a^2}{2a},\;\frac{2}{2a}$

$\implies\bf\;x=a,\;\displaystyle\frac{1}{a}$

$\therefore\textbf{The roots are a and \bf\displaystyle\frac{1}{a} }$

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