To determine which of the given quadratic equations has only one distinct solution, we can look for the equation that can be factored into a perfect square. A quadratic equation has only one distinct solution when the discriminant is zero, or when it can be expressed as \((x - a)^2 = 0\).
Let's evaluate each option:
-
\(x^2 - 5x + 4 = 0\)
This factors to \((x - 4)(x - 1) = 0\).
Solutions: \(x = 4\) and \(x = 1\). (Two distinct solutions) -
\(x^2 + 4x + 4 = 0\)
This factors to \((x + 2)(x + 2) = (x + 2)^2 = 0\).
Solution: \(x = -2\) (One distinct solution) -
\(x^2 + 5x + 4 = 0\)
This factors to \((x + 4)(x + 1) = 0\).
Solutions: \(x = -4\) and \(x = -1\). (Two distinct solutions) -
\(x^2 - 4 = 0\)
This can be factored as \((x - 2)(x + 2) = 0\).
Solutions: \(x = 2\) and \(x = -2\). (Two distinct solutions)
From the analysis, the equation with only one distinct solution is:
\(x^2 + 4x + 4 = 0\).