Question

Find the real and complex solution of the equation $x^2+6x+10=0$.

Short answer

The complex solutions are $x=-3+i,-3-i$.

Step-by-step solution

Step 1. Apply quadratic formula.

$\begin{array}{c}{x}^2+6x+10=\mathrm{0....}\left(\text{Given}\right)\\ \text{or,}x=\frac{-\left(6\right)\pm \sqrt{{\left(6\right)}^2-4\left(1\right)\left(10\right)}}{2\left(1\right)}\\ x=\frac{-\left(6\right)\pm \sqrt{-4}}{2}\mathrm{....}\left(1\right)\end{array}$

Step 2. By concept of complex numbers.

Further Solving for (1),

$\begin{array}{c}x=\frac{-6\pm 2i}{2}\mathrm{....}\left(\sqrt{-1}=i\right)\\ \text{or,}x=-3\pm i\\ x=-3+i,-3-i\end{array}$

Step 3. DESCRIPTION OF STEPS.

Since the roots are complex hence it has complex solution $x=-3+i,-3-i$.