In the following, determine whether the given quadratic equations have real roots and if so, find the roots:

$\sqrt2 x^2+7x+5\sqrt2=0$

Given:

Given quadratic equation is $\sqrt2 x^2+7x+5\sqrt2=0$.

To do:

We have to determine whether the given quadratic equation has real roots.

Solution:

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=\sqrt2, b=7$ and $c=5\sqrt2$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is

$D=b^2-4ac$.

Therefore,

$D=(7)^2-4(\sqrt2)(5\sqrt2)=49-20(2)=49-40=9$.

As $D>0$, the given quadratic equation has real roots and the roots are

$x=\frac{-b\pm \sqrt{D}}{2a}$

$x=\frac{-7\pm \sqrt{9}}{2(\sqrt2)}$ 

$x=\frac{-7\pm 3}{2\sqrt2}$ 

$x=\frac{-7+3}{2\sqrt2}$ or $x=\frac{-7-3}{2\sqrt2}$

$x=\frac{-4}{2\sqrt2}$ or $x=\frac{-10}{2\sqrt2}$

$x=\frac{-2}{\sqrt2}$ or $x=\frac{-5}{\sqrt2}$

$x=-\frac{\sqrt2 \times \sqrt2}{\sqrt2}$ or $x=-\frac{5}{\sqrt2}$

$x=-\sqrt{2}$ or $x=-\frac{5}{\sqrt2}$

The roots are $-\sqrt2$ and $-\frac{5}{\sqrt2}$. 

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Updated on: 10-Oct-2022

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