How many of the following are quadratic equation?
(i) $(x + 2)^{3} = 2 x (x^{2} - 1)$
(ii) $(x - 3) (2 x + 1) = x (x + 5)$
(iii) $x^{2} - 3 x + 2 = 0$

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Solution

The correct option is B $2$
(i) $(x + 2)^{3} = 2 x (x^{2} - 1)$ ........(1)
LHS = $(x + 2)^{3} = x^{3} + 3 \times x \times 2 (x + 2) + 2^{3}$
= $x^{3} + 6 x (x + 2) + 8$
= $x^{3} + 6 x^{2} + 12 x + 8$
RHS = $2 x (x^{2} - 1) = 2 x^{3} - 2 x$
Therefore, (1) can be written as
$x^{3} + 6 x^{2} + 12 x + 8 = 2 x^{3} - 2 x$
or $- x^{3} + 6 x^{2} + 14 x + 8 = 0$
or $x^{3} - 6 x^{2} - 14 x - 8 = 0$
It is a cubic equation and not a quadratic equation.

(ii) $(x - 3) (2 x + 1) = x (x + 5)$
$2 x^{2} + x - 6 x - 3 = x^{2} + 5 x$
$x^{2} - 5 x - 3 = 5 x$
$x^{2} - 10 x - 3 = 0$
Hence, the above is a quadratic equation.

(iii) $x^{2} - 3 x + 2 = 0$
is a quadratic equation.

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Similar questions

Check whether the following are quadratic equations :
(i) ${(x + 1)}^{2} = 2 (x - 3)$
(ii) $x^{2} - 2 x = (- 2) (3 - x)$
(iii) $(x - 2) (x + 1) = (x - 1) (x + 3)$
(iv) $(x - 3) (2 x + 1) = x (x + 5)$
(v) $(2 x - 1) (x - 3) = (x + 5) (x - 1)$
(vi) $x^{2} + 3 x + 1 = {(x - 2)}^{2}$
(vii) ${(x + 2)}^{3} = 2 x (x^{2} - 1)$
(viii) $x^{3} - 4 x^{2} - x + 1 = {(x - 2)}^{3}$