Question

Which of the following is a quadratic equation?

• A. $x^2+2x+1={(4-x)}^2+3$
• B. $-2x^2=(5-x)(2x-(\frac{2}{5}\left)\right)$
• C. $(k+1)x^2+\left(\frac{3}{2}\right)x=7,wherek=-1$
• D. $x^3-x^2={(x-1)}^3$

Answer 1

The correct option is D

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

Determine the quadratic equation.

The standard form of a quadratic equation is: $a{x}^2+bx+c=0,a\ne 0$

For point A)

$$\begin{gathered} x^2+2x+1={(4–x)}^2+3 \\ \Rightarrow x^2+2x+1=16–8x+x^2+3 \\ \Rightarrow 10x–18=0 \end{gathered}$$

It doesn't represent the standard form of a quadratic equation.

For point B)

$$\begin{gathered} -2x^2=(5–x)(2x–\frac{2}{5}) \\ \Rightarrow -2x^2=10x–2x^2–2+\frac{2}{5}x \\ \Rightarrow 52x–10=0 \end{gathered}$$

It doesn't represent the standard form of a quadratic equation.

For point C)

$$\begin{gathered} (k+1)x^2+\frac{3}{2}x=7,wherek=-1 \\ \Rightarrow (-1+1)x^2+\frac{3}{2}x=7 \\ \Rightarrow 3x–14=0 \end{gathered}$$

It doesn't represent the standard form of a quadratic equation.

For point D)

$$\begin{gathered} x^3–x^2={(x–1)}^3 \\ \Rightarrow x^3–x^2=x^3–3x^2+3x–1 \\ \Rightarrow 2x^2–3x+1=0 \end{gathered}$$

Hence, the above expression represents a quadratic equation. So$"x^3–x^2={(x–1)}^3"$ is the right answer

Hence, the correct answer is “Option D” that is $"x^3–x^2={(x–1)}^3"$.