State whether ${(x - 2)}^{2} + 1 = 2 x - 3$ is a quadratic equation.

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Solution

Step 1: Definition of a quadratic equation
The standard form of a quadratic equation is $a x^{2} + b x + c = 0$ .
For a given equation to be a quadratic equation, $a \neq 0$ and the highest power of the variable in the equation should be $2$ .
Step 2: To check whether the equation is a quadratic equation
Writing the given equation ${(x - 2)}^{2} + 1 = 2 x - 3$ in standard form.
$\begin{array}{rcl} {(x - 2)}^{2} + 1 & = & 2 x - 3 \\ \Rightarrow & x^{2} - 4 x + 4 + 1 = 2 x - 3 \\ \Rightarrow & x^{2} - 6 x + 8 = 0 \end{array}$
On comparing this equation with the standard form $a x^{2} + b x + c = 0$ , we can see that $a = 1$ and the highest power of the variable $x$ is $2$ .
Hence, the equation ${(x - 2)}^{2} + 1 = 2 x - 3$ is a quadratic equation.

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