Question

Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?
x^2 - 4 = 0
x^2 + 5x + 4 = 0
x^2 + 4x + 4 = 0
x^2 - 5x + 4 = 0

Answer 1

A quadratic equation has only one distinct solution when its discriminant is zero, which occurs when the equation can be expressed as a perfect square.

Let's analyze the equations one by one:

1. **Equation: \(x^2 - 4 = 0\)**
- This can be factored as \((x - 2)(x + 2) = 0\), which gives two distinct solutions: \(x = 2\) and \(x = -2\).

2. **Equation: \(x^2 + 5x + 4 = 0\)**
- This can be factored as \((x + 4)(x + 1) = 0\), which gives two distinct solutions: \(x = -4\) and \(x = -1\).

3. **Equation: \(x^2 + 4x + 4 = 0\)**
- This can be factored as \((x + 2)(x + 2) = 0\) or \((x + 2)^2 = 0\), which gives only one distinct solution: \(x = -2\).

4. **Equation: \(x^2 - 5x + 4 = 0\)**
- This can be factored as \((x - 4)(x - 1) = 0\), which gives two distinct solutions: \(x = 4\) and \(x = 1\).

From this analysis, the equation that has only one distinct solution is:

**\(x^2 + 4x + 4 = 0\)**.