Question

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

$x^2-6x+10=0$

Short answer

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

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 $a=1,b=-6,\text{\hspace{0.33em}and\hspace{0.33em}}c=10$.

Now use the quadratic formula to find the solutions as follows

role="math" localid="1644393679825" $$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{-\left(-6\right)\pm \sqrt{{\left(-6\right)}^2-4\left(1\right)\left(10\right)}}{2\left(1\right)} \\ =\frac{6\pm \sqrt{36-40}}{2} \\ =\frac{6\pm \sqrt{-4}}{2} \\ =\frac{6\pm 2i}{2} \\ =3\pm i \end{gathered}$$

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