Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$
The given equation is $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$.
Comparing it with $a x^{2}+b x+c=0$, we get
$a=\sqrt{2}, b=7$ and $c=5 \sqrt{2}$
$\therefore$ Discriminant, $D=b^{2}-4 a c=(7)^{2}-4 \times \sqrt{2} \times 5 \sqrt{2}=49-40=9>0$
So, the given equation has real roots.
Now, $\sqrt{D}=\sqrt{9}=3$
$\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-7+3}{2 \times \sqrt{2}}=\frac{-4}{2 \sqrt{2}}=-\sqrt{2}$
$\beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-7-3}{2 \times \sqrt{2}}=\frac{-10}{2 \sqrt{2}}=-\frac{5 \sqrt{2}}{2}$
Hence, $-\sqrt{2}$ and $-\frac{5 \sqrt{2}}{2}$ are the roots of the given equation.
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