Question

Find all solutions of the equation and express them in the form $a+bi$.

$x^2+3x+7=0$

Short answer

The solutions of the quadratic equation are $-\frac{3}{2}-\frac{\sqrt{19}}{2}i\text{and}-\frac{3}{2}+\frac{\sqrt{19}}{2}i$.

Step-by-step solution

Step 1. Concept of complex number.

A complex number is of the form $a+bi$. Where a and b are some real numbers and i is the imaginary number.

The number a is known as the real part of the complex number and b is known as imaginary part of the complex number.

Step 2. Concept of imaginary number.

The imaginary number i is defined as the square root of negative one. That is, $i=\sqrt{-1}$or $i^2=-1$.

The solutions of the quadratic equation $a{x}^2+bx+c=0$, where a, b, and c are constants, is given by the quadratic formula

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

Step 3. Solve the quadratic equation.

To find the solutions of the given quadratic equation, first compare it with the general quadratic equation$a{x}^2+bx+c=0$to get the value of coefficients $a=1,b=3,\text{\hspace{0.33em}and\hspace{0.33em}}c=7$.

Now use the quadratic formula to find the solutions as follows

localid="1644392609513" $$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{-\left(3\right)\pm \sqrt{{\left(3\right)}^2-4\left(1\right)\left(7\right)}}{2\left(1\right)} \\ =\frac{-3\pm \sqrt{9-28}}{2} \\ =\frac{-3\pm \sqrt{-19}}{2} \\ =\frac{-3\pm \sqrt{19}i}{2} \\ =-\frac{3}{2}\pm \frac{\sqrt{19}}{2}i \end{gathered}$$

Thus, the solutions of the quadratic equation are $-\frac{3}{2}-\frac{\sqrt{19}}{2}i\text{and}-\frac{3}{2}+\frac{\sqrt{19}}{2}i$.