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