State whether $(2 x + 1) (3 x + 2) = 6 (x - 1) (x - 2)$ 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 $(2 x + 1) (3 x + 2) = 6 (x - 1) (x - 2)$ in standard form.
$(2 x + 1) (3 x + 2) = 6 (x - 1) (x - 2) \Rightarrow 2 x (3 x + 2) + 1 (3 x + 2) = 6 [x (x - 2) - 1 (x - 2)] \Rightarrow 6 x^{2} + 4 x + 3 x + 2 = 6 (x^{2} - 2 x - x + 2) \Rightarrow 6 x^{2} + 7 x + 2 = 6 x^{2} - 18 x + 12 \Rightarrow 25 x - 10 = 0$
On comparing this equation with the standard form $a x^{2} + b x + c = 0$ , we can see that $a = 0$ and the highest power of the variable $x$ is $1$ .
Hence, the equation $(2 x + 1) (3 x + 2) = 6 (x - 1) (x - 2)$ is not a quadratic equation.

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