Which of the following is a quadratic equation?

Options

A. `x^(2)+2x+1=(4-x)^(2)+3`
B. `-2x^(2)=(5-x)(2x-2/5)`
C. `(k+1)x^(2)+3/2x=7`, where `k=-1`
D. `x^(3)-x^(2)=(x-1)^(3)`

Explanation

To determine which of the following is a quadratic equation, we need to understand the definition of a quadratic equation. A quadratic equation is an equation of the form:

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

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

Now, let's analyze the given options step by step:

### Step 1: Analyze the First Option
**First Option:**
\[ x^2 + 2x + 1 = 4 - x^2 + 3 \]

**Solution:**
1. Rearranging the equation:
\[ x^2 + 2x + 1 = 7 - x^2 \]
\[ x^2 + 2x + 1 + x^2 - 7 = 0 \]
\[ 2x^2 + 2x - 6 = 0 \]
\[ x^2 + x - 3 = 0 \] (after dividing by 2)

This is a quadratic equation since it has the form \( ax^2 + bx + c = 0 \) where \( a = 1 \), \( b = 1 \), and \( c = -3 \).

### Step 2: Analyze the Second Option
**Second Option:**
\[ -2x^2 = 5 - x(2x - 2/5) \]

**Solution:**
1. Simplifying the right side:
\[ -2x^2 = 5 - (2x^2 - 2/5x) \]
\[ -2x^2 = 5 - 2x^2 + 2/5x \]
\[ 0 = 5 + 2/5x \]

This does not contain an \( x^2 \) term after simplification, hence it is not a quadratic equation.

### Step 3: Analyze the Third Option
**Third Option:**
\[ (k + 1)x^2 + \frac{3}{2}x = 7 \text{ with } k = -1 \]

**Solution:**
1. Substituting \( k = -1 \):
\[ 0 \cdot x^2 + \frac{3}{2}x = 7 \]
\[ \frac{3}{2}x - 7 = 0 \]

This equation does not have an \( x^2 \) term, hence it is not a quadratic equation.

### Step 4: Analyze the Fourth Option
**Fourth Option:**
\[ x^3 - x^2 = (x - 1)^3 \]

**Solution:**
1. Expanding the right side:
\[ x^3 - x^2 = x^3 - 3x^2 + 3x - 1 \]
\[ -x^2 + 3x - 1 = 0 \]
\[ 2x^2 - 3x + 1 = 0 \]

This is a quadratic equation since it has the form \( ax^2 + bx + c = 0 \) where \( a = 2 \), \( b = -3 \), and \( c = 1 \).

### Conclusion
The first and fourth options are quadratic equations.