State whether $16 x^{2} - 3 = (2 x + 5) (5 x - 3)$ 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 $16 x^{2} - 3 = (2 x + 5) (5 x - 3)$ in standard form.
$16 x^{2} - 3 = (2 x + 5) (5 x - 3) \Rightarrow 16 x^{2} - 3 = 2 x (5 x - 3) + 5 (5 x - 3) \Rightarrow 16 x^{2} - 3 = 10 x^{2} - 6 x + 25 x - 15 \Rightarrow 6 x^{2} - 19 x + 12 = 0$
On comparing this equation with the standard form $a x^{2} + b x + c = 0$ , we can see that $a = 6$ and the highest power of the variable $x$ is $2$ .
Hence, the equation $16 x^{2} - 3 = (2 x + 5) (5 x - 3)$ is a quadratic equation.

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State whether $16 x^{2} - 3 = (2 x + 5) (5 x - 3)$ is a quadratic equation.