A quadratic equation has only one distinct solution when its discriminant is zero, which occurs when the equation can be expressed as a perfect square.
Let's analyze the equations one by one:
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Equation: \(x^2 - 4 = 0\)
- This can be factored as \((x - 2)(x + 2) = 0\), which gives two distinct solutions: \(x = 2\) and \(x = -2\).
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Equation: \(x^2 + 5x + 4 = 0\)
- This can be factored as \((x + 4)(x + 1) = 0\), which gives two distinct solutions: \(x = -4\) and \(x = -1\).
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Equation: \(x^2 + 4x + 4 = 0\)
- This can be factored as \((x + 2)(x + 2) = 0\) or \((x + 2)^2 = 0\), which gives only one distinct solution: \(x = -2\).
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Equation: \(x^2 - 5x + 4 = 0\)
- This can be factored as \((x - 4)(x - 1) = 0\), which gives two distinct solutions: \(x = 4\) and \(x = 1\).
From this analysis, the equation that has only one distinct solution is:
\(x^2 + 4x + 4 = 0\).