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Question

Which of the following is a quadratic equation?


A
x2+2x+1=(4x)2+3
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B
2x2=(5x)(2x25)
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C
(k+1)x2+32x=7    ( Where k =-1 )
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D
x3x2=(x1)3
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Solution

The correct option is D $$x^{3}- x^{2} = \left ( x - 1 \right )^{3}$$
The correct answer is option $$(D)$$
Main concept used : An equation of the form $$ax^{2} + bx + c = 0$$ where, $$a, b, c,$$ are numbers and $$a\neq0$$, is called a quadratic equation.

$$(a)$$ $$x^{2} + 2x + 1 = \left ( 4-x \right )^{2} + 3$$
$$\Rightarrow x^{2} + 2x + 1 = (4)^{2} + (x)^{2} -2(4) (x) + 3$$
$$\Rightarrow 2x + 1 = 16 - 8x + 3$$
$$\therefore$$ Coefficient of $$x^{2}$$ is zero or a = 0. So, it is not a quadratic equation.

(b) $$-2x^{2} = (5 - x) \left ( 2x - \dfrac{2}{5} \right )$$
$$\Rightarrow -2x^{2} = 10x - 2 -2x^{2} + \dfrac{2}{5}x$$
$$\Rightarrow -2x^{2} + 2x^{2} = 10x - 2 + \dfrac{2}{2}x$$
$$\Rightarrow 0 = 10x - 2 + \dfrac{2}{5}x$$
As the coefficient of $$x^{2}$$ in the above equation is zero or $$a = 0.$$
So, it is not a quadratic equation.

$$(c)$$ $$(k + 1)x^{2} + \dfrac{3}{2}x = 7$$ (where $$k = -1$$)
$$\Rightarrow (-1 + 1)x^{2} + \dfrac{3}{2}x = 7$$
So, the coefficient of $$x^{2}$$ is zero or $$a = 0$$. Hence, the equation in not quadratic.

(d) $$x^{3} - x^{2} = (x-1)^{3}$$
$$\Rightarrow x^{3} - x^{2} = (x)^{3} - (1)^{3} - 3 (x)^{2}(1) + 3(x) (1)^{2}$$
$$\Rightarrow x^{3} - x^{2} = x^{3} - 1 - 3x^{2} + 3x$$
$$\Rightarrow-x^{2} = -1 - 3x^{2} + 3x$$
$$\Rightarrow2x^{2} - 3x + 1 = 0$$
As the coefficient of $$x^{2}$$ in the above equation is $$2$$ or $$a = 2$$, so it is a quadratic equation.

Mathematics

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