Question

Which of the following is not a quadratic equation?

• A. \((x+2)(x-1)=(x+1)(x-3)\)
• B. \((x+1)^2=3(x-2)\)
• C. \((x+2)^2+3=x-1\)
• D. \((x+1)^3=x^3-2\)

Answer 1

The Correct Option is D

Step 1: Expand Each Given Equation

Option 1: \((x+1)^3 = x^3 - 2\) 

Expanding the left-hand side:

\((x+1)^3 = x^3 + 3x^2 + 3x + 1\)

So the equation becomes:

\(x^3 + 3x^2 + 3x + 1 = x^3 - 2\)

Subtracting \(x^3\) from both sides:

\(3x^2 + 3x + 1 = -2\)

\(3x^2 + 3x + 3 = 0\)

Since this contains \(x^3\) initially, it is not a quadratic equation.

Option 2: \((x+1)^2 = 3(x-2)\)

Expanding both sides:

\(x^2 + 2x + 1 = 3x - 6\)

Rearranging:

\(x^2 - x + 7 = 0\)

This is a quadratic equation.

Option 3: \((x+2)^2 + 3 = x - 1\)

Expanding:

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

Rearranging:

\(x^2 + 4x + 7 - x + 1 = 0\)

\(x^2 + 3x + 8 = 0\)

This is a quadratic equation.

Option 4: \((x+2)(x-1) = (x+1)(x-3)\)

Expanding both sides:

\(x^2 - x + 2x - 2 = x^2 - 3x + x - 3\)

\(x^2 + x - 2 = x^2 - 2x - 3\)

Cancel \(x^2\) from both sides:

\(x - 2 = -2x - 3\)

Rearrange:

\(x + 2x = -3 + 2\)

\(3x = -1\)

\(x = -\frac{1}{3}\)

This is a linear equation, not a quadratic equation.

Final Answer: \((x+1)^3 = x^3 - 2\)

Answer 2

To determine which equation is not a quadratic equation, we must recognize a quadratic equation typically takes the form \(ax^2+bx+c=0\), where \(a \neq 0\).

Let's analyze each option:

\((x+2)(x-1)=(x+1)(x-3)\)

Expanding both sides, \(x^2+x-2\) and \(x^2-2x-3\), respectively. Setting them equal gives \(x^2+x-2=x^2-2x-3\).

Simplifying, we get a common \(x^2\) term, leading to \(3x=1\), a linear equation. The expression equals zero, confirming it's a quadratic equation setup.

\((x+1)^2=3(x-2)\)

Expanding gives \(x^2+2x+1=3x-6\).

Rearranging all terms to one side, \(x^2-x+7=0\), shows it is in standard quadratic form.

\((x+2)^2+3=x-1\)

Expanding, we have \(x^2+4x+7=x-1\).

Rearranging, \(x^2+3x+8=0\), confirms a quadratic form.

\((x+1)^3=x^3-2\)

Expanding, we get \(x^3+3x^2+3x+1=x^3-2\).

Upon simplification, \(3x^2+3x+3=0\), clearly representing a cubic term.

This implies it is not a quadratic equation.

Therefore, the non-quadratic equation is \((x+1)^3=x^3-2\).