An equation has only one distinct solution (also known as a repeated solution) when the discriminant \( b^2 - 4ac \) is equal to zero.
Let's analyze each of the equations:
A. \( x^2 + 4x + 4 = 0 \)
- This factors to \( (x + 2)(x + 2) = 0 \).
- The solution is \( x = -2 \) (one distinct solution).
B. \( x^2 - 5x + 4 = 0 \)
- This factors to \( (x - 1)(x - 4) = 0 \).
- The solutions are \( x = 1 \) and \( x = 4 \) (two distinct solutions).
C. \( x^2 - 4 = 0 \)
- This factors to \( (x - 2)(x + 2) = 0 \).
- The solutions are \( x = 2 \) and \( x = -2 \) (two distinct solutions).
D. \( x^2 + 5x + 4 = 0 \)
- This factors to \( (x + 1)(x + 4) = 0 \).
- The solutions are \( x = -1 \) and \( x = -4 \) (two distinct solutions).
The only equation that has one distinct solution is:
A. \( x^2 + 4x + 4 = 0 \).