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Choose the correct answer from the given four options in the following questions:
Which of the following is a quadratic equation?
(A) \( x^{2}+2 x+1=(4-x)^{2}+3 \)
(B) \( -2 x^{2}=(5-x)\left(2 x-\frac{2}{5}\right) \)
(C) \( (k+1) x^{2}+\frac{3}{2} x=7 \), where \( k=-1 \)
(D) \( x^{3}-x^{2}=(x-1)^{3} \)
To do:
We have to find the correct answer.
Solution:
$x^{2}+2 x+1=(4-x)^{2}+3$
$x^{2}+2 x+1=16+x^{2}-8x+3$
$10x-18=0$ which is not of the form $a x^{2}+b x+c, a≠0$.
Therefore, the equation $x^{2}+2 x+1=(4-x)^{2}+3$ is not a quadratic equation.
$-2 x^{2} =(5-x)(2 x-\frac{2}{5})$
$-2 x^{2}=10 x-2 x^{2}-2+\frac{2 x}{5}$
$50 x+2 x-10=0$
$52 x-10=0$ which is also not a quadratic equation.
$x^{2}(k+1)+\frac{3}{2} x=7$
$k=-1$
$x^{2}(-1+1)+\frac{3}{2} x=7$
$3 x-14=0$ which is also not a quadratic equation.
$x^{3}-x^{2}=(x-1)^{3}$
$x^{3}-x^{2}=x^{3}-3 x^{2}(1)+3 x(1)^{2}-(1)^{3}$
$x^{3}-x^{2} =x^{3}-3 x^{2}+3 x-1$
$-x^{2}+3 x^{2}-3 x+1 =0$
$2 x^{2}-3 x+1=0$ which represents a quadratic equation (it is of the quadratic form $a x^{2}+b x+c=0, a≠0$)