Question

Solve by using the Quadratic Formula:$4a^2-2a+8=0$

Short answer

The solution of the quadratic equation is$a=\frac{1\pm \sqrt{31}i}{4}$

Step-by-step solution

Given information

We are given a quadratic equation$4a^2-2a+8=0$

Identify the values of a,b,c

On comparing with standard equation we get the values as$x=4,y=-2,z=8$

Write quadratic formula and substitute the values of x,y,z

The quadratic formula can be given as

$a=\frac{-y\pm \sqrt{y^2-4xz}}{2x}$

Substituting the values we get

$a=\frac{2\pm \sqrt{4-4\left(4\right)\left(8\right)}}{2\left(4\right)}$

Simplify and also find the common factor and remove it

On simplifying we get

$$\begin{gathered} a=\frac{2\pm \sqrt{4-128}}{8} \\ a=\frac{2\pm \sqrt{-124}}{8} \\ a=\frac{2\pm \sqrt{124}i}{8} \end{gathered}$$

Now we find and remove the common factor

$$\begin{gathered} a=\frac{2\pm 2\sqrt{31}i}{8} \\ a=\frac{2(1+\sqrt{31}i)}{8} \\ a=\frac{1+\sqrt{31}i}{4} \end{gathered}$$

Rewrite in standard form

It can be written in standard form as$a=\frac{1}{4}\pm \frac{\sqrt{31}i}{4}$

Conclusion

The solution of the above quadratic equation is$a=\frac{1}{4}\pm \frac{\sqrt{31}i}{4}$