An equation has only one distinct solution when its discriminant is equal to zero. The discriminant for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by the formula \( D = b^2 - 4ac \).
Let's calculate the discriminant for each equation:
A: \( x^2 + 4x + 4 = 0 \)
- Here, \( a = 1 \), \( b = 4 \), \( c = 4 \).
- \( D = 4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0 \).
- This equation has only one distinct solution.
B: \( x^2 + 5x + 4 = 0 \)
- Here, \( a = 1 \), \( b = 5 \), \( c = 4 \).
- \( D = 5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9 \).
- This equation has two distinct solutions.
C: \( x^2 - 4 = 0 \)
- This can be rewritten as \( x^2 + 0x - 4 = 0 \), so \( a = 1 \), \( b = 0 \), \( c = -4 \).
- \( D = 0^2 - 4 \cdot 1 \cdot (-4) = 0 + 16 = 16 \).
- This equation has two distinct solutions.
D: \( x^2 - 5x + 4 = 0 \)
- Here, \( a = 1 \), \( b = -5 \), \( c = 4 \).
- \( D = (-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9 \).
- This equation has two distinct solutions.
In conclusion, the equation that has only one distinct solution is:
A: \( x^2 + 4x + 4 = 0 \).
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