To determine which quadratic equation has only one distinct solution, we can look for a situation where the quadratic can be factored into a perfect square. This occurs when the discriminant (b² - 4ac) is zero.
Let's analyze each equation:
-
x² - 5x + 4 = 0
Factors: (x - 1)(x - 4) = 0
Solutions: x = 1, x = 4 (two distinct solutions) -
x² - 4 = 0
Factors: (x - 2)(x + 2) = 0
Solutions: x = 2, x = -2 (two distinct solutions) -
x² + 4x + 4 = 0
Factors: (x + 2)(x + 2) = 0 (or (x + 2)² = 0)
Solutions: x = -2 (one distinct solution) -
x² + 5x + 4 = 0
Factors: (x + 1)(x + 4) = 0
Solutions: x = -1, x = -4 (two distinct solutions)
From this analysis, the equation that has only one distinct solution is:
x² + 4x + 4 = 0 (which factors to (x + 2)² = 0, yielding the double root x = -2).