Which of the following is not a quadratic equation ?

Options

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

Answer

To determine which of the given options is not a quadratic equation, we first need to understand the definition of a quadratic equation. A quadratic equation is typically in the form of:

\[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

Now, let's analyze each option step by step.

### Step 1: Analyze Option 1
**Equation:**
\[ 2(x - 1)^2 = 4x^2 - 2x + 1 \]

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

2. Rearranging gives:
\[ 2x^2 - 4x + 2 - 4x^2 + 2x - 1 = 0 \]
\[ -2x^2 - 2x + 1 = 0 \]
\[ 2x^2 + 2x - 1 = 0 \]

This is a quadratic equation.

### Step 2: Analyze Option 2
**Equation:**
\[ 2x - x^2 = x^2 + 5 \]

**Solution:**
1. Rearranging gives:
\[ 2x - x^2 - x^2 - 5 = 0 \]
\[ -2x^2 + 2x - 5 = 0 \]
\[ 2x^2 - 2x + 5 = 0 \]

This is also a quadratic equation.

### Step 3: Analyze Option 3
**Equation:**
\[ \sqrt{2x + \sqrt{3}}^2 + x^2 = 3x^2 - 5x \]

**Solution:**
1. Expand the left side:
\[ 2x + 3 + x^2 = 3x^2 - 5x \]

2. Rearranging gives:
\[ x^2 + 2x + 3 - 3x^2 + 5x = 0 \]
\[ -2x^2 + 7x + 3 = 0 \]

This is a quadratic equation.

### Step 4: Analyze Option 4
**Equation:**
\[ (x^2 + 2x)^2 = x^4 + 3 + 4x^3 \]

**Solution:**
1. Expand the left side:
\[ x^4 + 4x^3 + 4x^2 = x^4 + 3 + 4x^3 \]

2. Rearranging gives:
\[ 4x^2 - 3 = 0 \]

This is also a quadratic equation.

### Conclusion
After analyzing all the options, we find that all options except for Option 3 yield quadratic equations.

**Final Answer:**
Option 3 is not a quadratic equation.