Which of the following is not a Quadratic Equation ?

Options

A. `2 (x-1) ^(2) = 4x ^(2) - 2x +1`
B. `3x-x ^(2) =x ^(2) +6`
C. `(sqrt3 x + sqrt2)^(2) = 2x ^(2) - 5x`
D. `(x ^(2) + 2x )^(2) = x ^(4) + 3+ 4x ^(2)`

Answer

To determine which of the given equations is not a quadratic equation, we need to understand the standard form of a quadratic equation. The standard form is:

\[ Ax^2 + Bx + C = 0 \]

where:
- \( A \), \( B \), and \( C \) are constants,
- \( x \) is the variable,
- The highest degree of \( x \) must be 2,
- \( A \) cannot be equal to 0.

Now, let's analyze each of the provided equations step by step.

### Step 1: Analyze the first equation
**Equation:** \( 2(x - 1)^2 = 4x^2 - 2x + 1 \)

1. Expand the left side:
\[ 2(x - 1)^2 = 2(x^2 - 2x + 1) = 2x^2 - 4x + 2 \]

2. Set the equation to zero:
\[ 2x^2 - 4x + 2 - (4x^2 - 2x + 1) = 0 \]
Simplifying gives:
\[ -2x^2 + 6x + 1 = 0 \]

3. The highest degree is 2, so this is a quadratic equation.

### Step 2: Analyze the second equation
**Equation:** \( 3x - x^2 = x^2 + 6 \)

1. Rearranging gives:
\[ -x^2 - 6 + 3x = 0 \implies -2x^2 + 3x - 6 = 0 \]

2. The highest degree is 2, so this is also a quadratic equation.

### Step 3: Analyze the third equation
**Equation:** \( \sqrt{3}x + \sqrt{2}^2 = 2x^2 - 5x \)

1. Rearranging gives:
\[ 3x + 2 = 2x^2 - 5x \]
Rearranging further yields:
\[ 2x^2 - 8x - 2 = 0 \]

2. The highest degree is 2, so this is a quadratic equation.

### Step 4: Analyze the fourth equation
**Equation:** \( (x^2 + 2x)^2 = 4 + 2x \)

1. Expanding the left side:
\[ (x^2 + 2x)^2 = x^4 + 4x^3 + 4x^2 \]

2. Rearranging gives:
\[ x^4 + 4x^3 + 4x^2 - (4 + 2x) = 0 \]
Simplifying yields:
\[ x^4 + 4x^3 + 4x^2 - 2x - 4 = 0 \]

3. The highest degree is 4, so this is **not** a quadratic equation.

### Conclusion
The equation that is not a quadratic equation is the fourth one.

**Final Answer:** The fourth equation is not a quadratic equation.

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