To determine which quadratic equation has only one distinct solution, we can look for a perfect square trinomial or check if the discriminant (b² - 4ac) is zero.
We can analyze each equation:
A) \(x^2 + 5x + 4 = 0\)
This factors to \((x + 1)(x + 4) = 0\)
Solutions: \(x = -1\) and \(x = -4\) (two distinct solutions)
B) \(x^2 + 4x + 4 = 0\)
This factors to \((x + 2)(x + 2) = 0\)
Solution: \(x = -2\) (one distinct solution)
C) \(x^2 - 4 = 0\)
This factors to \((x - 2)(x + 2) = 0\)
Solutions: \(x = 2\) and \(x = -2\) (two distinct solutions)
D) \(x^2 - 5x + 4 = 0\)
This factors to \((x - 1)(x - 4) = 0\)
Solutions: \(x = 1\) and \(x = 4\) (two distinct solutions)
From this analysis, the equation that has only one distinct solution is: B) \(x^2 + 4x + 4 = 0\).