Check whether the following is a quadratic equation:
$x^{3} - {4 x}^{2} - x + 1 = {(x - 2)}^{3}$

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Solution

$x^{3} - 4 x^{2} - x + 1 = {(x - 2)}^{3}$
$\Rightarrow x^{3} - 4 x^{2} - x + 1 = x^{3} - 8 - 6 x (x - 2)$
$\Rightarrow - 4 x^{2} - x + 1 = - 8 - 6 x^{2} + 12 x$
$\Rightarrow - 4 x^{2} - x + 1 + 8 + 6 x^{2} - 12 x = 0$
$\Rightarrow 2 x^{2} - 13 x + 9 = 0$

The above-obtained equation has its highest power $2$ hence, it is a quadratic equation.

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I s

x^{3} - 4 x^{2} - x + 1 = (x - 2)^{3}

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