To determine which of the given quadratic equations has only one distinct solution, we can check the discriminant of each equation. The discriminant is given by the formula:
\[ D = b^2 - 4ac \]
A quadratic equation \( ax^2 + bx + c = 0 \) has only one distinct solution if the discriminant \( D = 0 \).
Let's analyze each of the equations:
-
\( x^2 - 5x + 4 = 0 \)
- Here, \( a = 1 \), \( b = -5 \), \( c = 4 \)
- Discriminant: \( D = (-5)^2 - 4(1)(4) = 25 - 16 = 9 \) (not zero)
-
\( x^2 - 4 = 0 \)
- This can be rewritten as \( x^2 - 0x - 4 = 0 \) with \( a = 1 \), \( b = 0 \), \( c = -4 \)
- Discriminant: \( D = 0^2 - 4(1)(-4) = 0 + 16 = 16 \) (not zero)
-
\( x^2 + 5x + 4 = 0 \)
- Here, \( a = 1 \), \( b = 5 \), \( c = 4 \)
- Discriminant: \( D = 5^2 - 4(1)(4) = 25 - 16 = 9 \) (not zero)
-
\( x^2 + 4x + 4 = 0 \)
- Here, \( a = 1 \), \( b = 4 \), \( c = 4 \)
- Discriminant: \( D = 4^2 - 4(1)(4) = 16 - 16 = 0 \) (zero)
From our analysis, the equation that has only one distinct solution is:
\( x^2 + 4x + 4 = 0 \).