College Algebra, Chapter 2, 2.3, Section 2.3, Problem 38

Solve the equation $x^2 + 3 = 2x$ algebraically and graphically.

$\begin{equation} \begin{aligned} x^2 + 3 &= 2x\\ \\ x^2 - 2x + 3 &= 0 && \text{Subtract } 2x \end{aligned} \end{equation}$

Graphically, the $x$-intercept does not exist




By solving the exact value,

$\begin{equation} \begin{aligned} x^2 - 2x + 3 &= 0 \\ \\ x^2 - 2x &= -3 && \text{Subtract } 3\\ \\ x^2 - 2x + 1 &= -3 + 1 && \text{Complete the square: Add } \left( \frac{-2}{2} \right)^2 = 1\\ \\ (x - 1)^2 &= -2 && \text{Perfect Square}\\ \\ x -1 &= \pm \sqrt{-2} && \text{Take the square root}\\ \\ x &= 1 \pm \sqrt{-2} && \text{Add } 1 \end{aligned} \end{equation}$

We see that $x$ is an imaginary number, thus, $x$-intercept does not exist