Find the roots of the quadratic equations by using the quadratic formula in each of the following:
$ \frac{1}{2} x^{2}-\sqrt{11} x+1=0 $

Given:

Given quadratic equation is \( \frac{1}{2} x^{2}-\sqrt{11} x+1=0 \).

To do:

We have to find the roots of the given quadratic equation.

Solution:

\( \frac{1}{2} x^{2}-\sqrt{11} x+1=0 \)

The above equation is of the form $ax^2 + bx + c = 0$, where $a = \frac{1}{2}, b = -\sqrt{11}$ and $c =1$

Discriminant $\mathrm{D} =b^{2}-4 a c$

$=(-\sqrt{11})^{2}-4 \times \frac{1}{2} \times 1$

$=11-2$

$=9$

$\mathrm{D}>0$

Let the roots of the equation are $\alpha$ and $\beta$

$\alpha =\frac{-b+\sqrt{\mathrm{D}}}{2 a}$

$=\frac{-(-\sqrt{11})+\sqrt{9}}{2(\frac{1}{2})}$

$=\frac{\sqrt{11}+3}{1}$

$=3+\sqrt{11}$

$\beta =\frac{-b-\sqrt{\mathrm{D}}}{2 a}$

$=\frac{-(-\sqrt{11})-\sqrt{9}}{2(\frac{1}{2})}$

$=\frac{\sqrt{11}-3}{1}$

$=-3+\sqrt{11}$

Hence, the roots of the given quadratic equation are $3+\sqrt{11}, -3+\sqrt{11}$. 

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Updated on: 10-Oct-2022

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