Question

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

$2x^2-2x+1=0$

Short answer

The solutions of the quadratic equation are $\frac{1}{2}-\frac{1}{2}i\text{\hspace{0.33em}\hspace{0.33em}and}\frac{1}{2}+\frac{1}{2}i$.

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 $a=2,b=-2,\text{\hspace{0.33em}and\hspace{0.33em}}c=1$.

Now use the quadratic formula to find the solutions as follows

role="math" localid="1644398756383" $$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{-\left(-2\right)\pm \sqrt{{\left(-2\right)}^2-4\left(2\right)\left(1\right)}}{2\left(2\right)} \\ =\frac{2\pm \sqrt{4-8}}{4} \\ =\frac{2\pm \sqrt{-4}}{4} \\ =\frac{2\pm 2i}{4} \\ =\frac{1}{2}\pm \frac{1}{2}i \end{gathered}$$

Thus, the solutions of the quadratic equation are $\frac{1}{2}-\frac{1}{2}i\text{\hspace{0.33em}\hspace{0.33em}and}\frac{1}{2}+\frac{1}{2}i$.