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Find the roots of the quadratic equations by using the quadratic formula in each of the following:
\( x^{2}+2 \sqrt{2} x-6=0 \)
Given:
Given quadratic equation is \( x^{2}+2 \sqrt{2} x-6=0 \).
To do:
We have to find the roots of the given quadratic equation.
Solution:
\( x^{2}+2 \sqrt{2} x-6=0 \)
The above equation is of the form $ax^2 + bx + c = 0$, where $a = 1, b = 2 \sqrt{2}$ and $c =-6$
Discriminant $\mathrm{D} =b^{2}-4 a c$
$=(2 \sqrt{2})^{2}-4 \times (1)\times(-6)$
$=8+24$
$=32$
$\mathrm{D}>0$
Let the roots of the equation are $\alpha$ and $\beta$
$\alpha =\frac{-b+\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-2 \sqrt{2}+\sqrt{32}}{2(1)}$
$=\frac{-2 \sqrt{2}+4\sqrt2}{2}$
$=\frac{2\sqrt2}{2}$
$=\sqrt2$
$\beta =\frac{-b-\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-2 \sqrt{2}-\sqrt{32}}{2(1)}$
$=\frac{-2 \sqrt{2}-4\sqrt{2}}{2}$
$=\frac{-6\sqrt{2}}{2}$
$=-3\sqrt2$
Hence, the roots of the given quadratic equation are $\sqrt2, -3\sqrt2$.