Question

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

$2x^2+x-4=0$

Answer 1

The given equation is $2x^2+x-4=0$.

Comparing it with $a{x}^2+bx+c=0$, we get

a = 2, b = 1 and c = −4

∴ Discriminant, D = $b^2-4ac={\left(1\right)}^2-4\times 2\times \left(-4\right)=1+32=33>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{33}$

$$\begin{gathered} \therefore \alpha =\frac{-b+\sqrt{D}}{2a}=\frac{-1+\sqrt{33}}{2\times 2}=\frac{-1+\sqrt{33}}{4} \\ \beta =\frac{-b-\sqrt{D}}{2a}=\frac{-1-\sqrt{33}}{2\times 2}=\frac{-1-\sqrt{33}}{4} \end{gathered}$$

Hence, $\frac{-1+\sqrt{33}}{4}$ and $\frac{-1-\sqrt{33}}{4}$ are the roots of the given equation.