A quadratic equation has only one distinct solution when it can be factored into a perfect square. This occurs when the discriminant (\(b^2 - 4ac\)) is equal to zero.
Let's analyze each of the given equations:
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\(x^2 + 5x + 4 = 0\)
- Discriminant: \(5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9\) (two distinct solutions)
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\(x^2 + 4x + 4 = 0\)
- Discriminant: \(4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0\) (one distinct solution)
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\(x^2 - 4 = 0\)
- Discriminant: \(0^2 - 4 \cdot 1 \cdot (-4) = 0 + 16 = 16\) (two distinct solutions)
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\(x^2 - 5x + 4 = 0\)
- Discriminant: \((-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9\) (two distinct solutions)
From this analysis, the equation that has only one distinct solution is:
\(x^2 + 4x + 4 = 0\)