Question

Write the discriminant of $\sqrt{5}{x}^2-3\sqrt{6}x-\sqrt{20}=0$ and determine the nature of its root.

Answer 1

Step 1: Comparison with the standard form of quadratic equation $a{x}^2+bx+c=0$

The given quadratic equation is $\sqrt{5}{x}^2-3\sqrt{6}x-\sqrt{20}=0$.

Here, $a=\sqrt{5},b=-3\sqrt{6}$ and $c=-\sqrt{20}$.

Step 2: Finding the discriminant

Using the formula for discriminant$\left(D\right)=b^2-4ac$ and substituting the values of $a,b$ and $c$ we get,

$\begin{array}{rcl}D& =& {(-3\sqrt{6})}^2-4\left(\sqrt{5}\right)(-\sqrt{20})\\ & =& 54+4\sqrt{100}\\ & =& 54+40\\ & =& 94\end{array}$.

Step 3: Determining the nature of the roots of the equation

We know that the value of the discriminant decides the nature of the roots of the quadratic equation.

If $D>0$, then the quadratic equation has real and unequal roots.

If $D=0$, then the quadratic equation has real and equal roots.

If $D<0$, then the quadratic equation has no real roots.

For the given quadratic equation $D>0$. So, the equation has real and unequal roots.

Therefore, the quadratic equation $\sqrt{5}{x}^2-3\sqrt{6}x-\sqrt{20}=0$ has real and unequal roots.