A quadratic equation has only one distinct solution when it can be expressed as a perfect square trinomial or when the discriminant (b^2 - 4ac) is zero.
Let's analyze the provided options:
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x² − 5x + 4 = 0: This factors to (x - 4)(x - 1) = 0. The solutions are x = 4 and x = 1 (two distinct solutions).
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x² + 4x + 4 = 0: This factors to (x + 2)(x + 2) = 0, which is a perfect square. The solution is x = -2 (one distinct solution).
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x² + 5x + 4 = 0: This factors to (x + 4)(x + 1) = 0. The solutions are x = -4 and x = -1 (two distinct solutions).
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x² − 4 = 0: This factors to (x - 2)(x + 2) = 0. The solutions are x = 2 and x = -2 (two distinct solutions).
The option that has only one distinct solution is:
x² + 4x + 4 = 0