State whether $x (2 x + 3) = x^{2} + 1$ 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 $x (2 x + 3) = x^{2} + 1$ in standard form.
$x (2 x + 3) = x^{2} + 1 \Rightarrow 2 x^{2} + 3 x = x^{2} + 1 \Rightarrow x^{2} + 3 x - 1 = 0$
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 $x (2 x + 3) = x^{2} + 1$ is a quadratic equation.

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