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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 $3+\sqrt{11}, -3+\sqrt{11}$.