  Question

Which of the following is a quadratic equation?

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