Question

In the following exercises, solve by using the Quadratic Formula:$2a^2-6a+3=0$

Short answer

The solution of the quadratic equation is$a=\frac{3\pm \sqrt{3}}{2}$

Step-by-step solution

Given information

We are given a quadratic equation$2a^2-6a+3=0$

Identify the values of x,y,z

On comparing with standard equation we get$x=2,y=-6,z=3$

Write the quadratic formula and substitute the values in it

The quadratic formula can be given as

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

On substituting the values we get

$a=\frac{6\pm \sqrt{36-4\left(2\right)\left(3\right)}}{2\left(2\right)}$

Simplify

On simplifying we get,

$$\begin{gathered} a=\frac{6\pm \sqrt{36-24}}{4} \\ a=\frac{6\pm \sqrt{12}}{4} \\ a=\frac{6\pm 2\sqrt{3}}{4} \end{gathered}$$

Take out and remove the common factor

We get

$$\begin{gathered} a=\frac{2(3\pm \sqrt{3})}{4} \\ a=\frac{3\pm \sqrt{3}}{2} \end{gathered}$$

Conclusion

The solution of the quadratic equation of$a=\frac{3\pm \sqrt{3}}{2}$