Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?
A. x^2 + 4x + 4=0
B. X^2-4=0
C. X^2-5x+4=0
D. X^2 + 5x + 4=0
The quadratic equation that has only one distinct solution can be found by looking at the discriminant. The discriminant is the term inside the square root in the quadratic formula and can be calculated as b^2 - 4ac.
A. x^2 + 4x + 4 = 0
In this equation, a = 1, b = 4, and c = 4.
The discriminant is b^2 - 4ac = 4^2 - 4(1)(4) = 16 - 16 = 0.
Since the discriminant is zero, this equation has one distinct solution.
B. X^2-4 = 0
In this equation, a = 1, b = 0, and c = -4.
The discriminant is b^2 - 4ac = 0^2 - 4(1)(-4) = 16.
Since the discriminant is positive, this equation has two distinct solutions.
C. X^2-5x+4 = 0
In this equation, a = 1, b = -5, and c = 4.
The discriminant is b^2 - 4ac = (-5)^2 - 4(1)(4) = 25 - 16 = 9.
Since the discriminant is positive, this equation has two distinct solutions.
D. X^2 + 5x + 4 = 0
In this equation, a = 1, b = 5, and c = 4.
The discriminant is b^2 - 4ac = 5^2 - 4(1)(4) = 25 - 16 = 9.
Since the discriminant is positive, this equation has two distinct solutions.
Therefore, the quadratic equation that has only one distinct solution is option A, x^2 + 4x + 4 = 0.