To determine which equation has only one distinct solution, we can check the discriminant of each quadratic equation. The discriminant (\(D\)) is given by the formula:
\[ D = b^2 - 4ac \]
A quadratic equation has only one distinct solution when the discriminant is equal to 0 (\(D = 0\)).
Let's evaluate each equation:
a) \(x^2 + 4x + 4 = 0\)
Here, \(a = 1\), \(b = 4\), and \(c = 4\).
\[ D = 4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0 \]
This equation has only one distinct solution.
b) \(x^2 + 5x + 4 = 0\)
Here, \(a = 1\), \(b = 5\), and \(c = 4\).
\[ D = 5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9 \]
This equation has two distinct solutions.
c) \(x^2 - 4 = 0\)
Here, \(a = 1\), \(b = 0\), and \(c = -4\).
\[ D = 0^2 - 4 \cdot 1 \cdot (-4) = 0 + 16 = 16 \]
This equation also has two distinct solutions.
d) \(x^2 - 5x + 4 = 0\)
Here, \(a = 1\), \(b = -5\), and \(c = 4\).
\[ D = (-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9 \]
This equation has two distinct solutions.
The only equation that has one distinct solution is:
a) \(x^2 + 4x + 4 = 0\).
1 answer