Question

Write the discriminant of $\sqrt{2}{x}^2-3\sqrt{5}x+\sqrt{18}=0$ 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 $\sqrt{2}{x}^2-3\sqrt{5}x+\sqrt{18}=0$.

Here, $a=\sqrt{2},b=-3\sqrt{5}$ and $c=\sqrt{18}$.

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,

$$\begin{gathered} \begin{array}{rcl}& & D={(-3\sqrt{5})}^2-4\left(\sqrt{2)}\right(\sqrt{18})\\ & & =45-4\left(6\right) \\ =45-24 \\ =21\end{array} \end{gathered}$$

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 $\sqrt{2}{x}^2-3\sqrt{5}x+\sqrt{18}=0$ has real and unequal roots.