Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

Question:

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

$\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$

Solution:

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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