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Which of the following is a quadratic equation?

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

- 2 x^{2} = (5 - x) (2 x - \frac{2}{5})

(k + 1) x^{2} + \frac{3}{2} x = 7

( Where k =-1)

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

answer is D.

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Detailed Solution

It is given that we have to identify the quadratic equation from the given options.
We know that an equation of the form

a x^{2} + bx + c = 0

where,

a, b, c,

are numbers and

a \neq 0

, is called a quadratic equation.

(1)

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

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

\Rightarrow 2 x + 1 = 16 - 8 x + 3

∴

Coefficient of

x^{2}

is zero or a = 0.
So, it is not a quadratic equation.
(2)

- 2 x^{2} = (5 - x) (2 x - \frac{2}{5})

\Rightarrow - 2 x^{2} = 10 x - 2 - 2 x^{2} + \frac{2}{5} x

\Rightarrow - 2 x^{2} + 2 x^{2} = 10 x - 2 + \frac{2}{2} x

\Rightarrow 0 = 10 x - 2 + \frac{2}{5} x

As the coefficient of

x^{2}

in the above equation is zero or

a = 0 .

So, it is not a quadratic equation.

(3)

(k + 1) x^{2} + \frac{3}{2} x = 7

(Where

k = - 1

)

\Rightarrow (- 1 + 1) x^{2} + \frac{3}{2} x = 7

So, the coefficient of

x^{2}

is zero or

a = 0

.
Hence, the equation in not quadratic.
(4)

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

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

\Rightarrow x^{3} - x^{2} = x^{3} - 1 - 3 x^{2} + 3 x

\Rightarrow - x^{2} = - 1 - 3 x^{2} + 3 x

\Rightarrow 2 x^{2} - 3 x + 1 = 0

As the coefficient of

x^{2}

in the above equation is

2

a = 2

, so it is a quadratic equation.
Hence, the correct option is 4.

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