Question

State whether the following quadratic equations have two distinct real roots. Justify your answer.

$\sqrt{2}{x}^2-\frac{3}{\sqrt{2}}x+\frac{1}{\sqrt{2}}=0$

Answer 1

Nature of roots:

The discriminants of a quadratic equation $a{x}^2+bx+c=0$ is given by:

$D=b^2-4ac$

• If $D>0$, the roots are distinct and real.
• If $D=0$, the roots are equal and real.
• If $D<0$, the roots will be imaginary.

The discriminant of the given quadratic equation $\sqrt{2}{x}^2-\frac{3}{\sqrt{2}}x+\frac{1}{\sqrt{2}}=0$is computed as:

$$\begin{gathered} \therefore D={\left(-\frac{3}{\sqrt{2}}\right)}^2-4·\sqrt{2}·\frac{1}{\sqrt{2}} \\ \Rightarrow D=\frac{9}{2}-4 \\ \Rightarrow D=\frac{1}{2} \\ \Rightarrow D>0 \end{gathered}$$

Since the discriminant is greater than zero.

Hence, the roots are distinct and real.