Question

In the following exercises, solve by using the Quadratic Formula.
$2x^2+3x+3=0$

Short answer

The solutions of the given equation are $x=-\frac{3}{4}\pm \frac{\sqrt{15}}{4}i$.

Step-by-step solution

Step 1. Given information.

The given equation is:

$2x^2+3x+3=0$

Step 2. Substitute the values in Quadratic Formula.

Quadratic Formula for the equation $a{x}^2+bx+c=0$is:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

In the given equation, $a=2,b=3,c=3$. Substitute these values in Quadratic Formula.

$x=\frac{-3\pm \sqrt{3^2-4\left(2\right)\left(3\right)}}{2\left(2\right)}$

Step 3. Simplify the value.

On simplification, we get

$\begin{array}{rcl}x& =& \frac{-3\pm \sqrt{9-24}}{4}\\ x& =& \frac{-3\pm \sqrt{-15}}{4}\\ x& =& \frac{-3}{4}\pm \frac{\sqrt{-15}}{4}\\ x& =& -\frac{3}{4}\pm \frac{\sqrt{15}}{4}i[\because \sqrt{-1}=i]\end{array}$

Step 4. Conclusion.

The solutions of the given equation are $x=-\frac{3}{4}\pm \frac{\sqrt{15}}{4}i$.