Question

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

$3x^2+1=0$

Short answer

The solutions of the quadratic equation are $0-\frac{1}{\sqrt{3}}i\text{and}0+\frac{1}{\sqrt{3}}i$.

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 $a=3,b=0,\text{\hspace{0.33em}and\hspace{0.33em}}c=1$.

Now use the quadratic formula to find the solutions as follows

role="math" localid="1644329095944" $$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{-0\pm \sqrt{{\left(0\right)}^2-4\left(3\right)\left(1\right)}}{2\left(3\right)} \\ =\frac{\pm \sqrt{0-12}}{6} \\ =\pm \frac{\sqrt{12}i}{6} \\ =\pm \frac{2\sqrt{3}i}{6} \\ =\pm \frac{\sqrt{3}i}{3} \\ =\pm \frac{1}{\sqrt{3}}i \end{gathered}$$

Thus, the solutions of the quadratic equation are $0-\frac{1}{\sqrt{3}}i\text{and}0+\frac{1}{\sqrt{3}}i$.