Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point)

Question

Responses

x2+4x+4=0
x squared plus 4 x plus 4 equals 0

x2+5x+4=0
x squared plus 5 x plus 4 equals 0

x2−5x+4=0
x squared minus 5 x plus 4 equals 0

x2−4=0

Answer 1

An equation has only one distinct solution when it can be factored into a perfect square. This happens when the discriminant (b² - 4ac) equals zero.

Let's analyze each equation:

1. \(x^2 + 4x + 4 = 0\):

   • This factors to \((x + 2)^2 = 0\).
   • It has one distinct solution: \(x = -2\).

2. \(x^2 + 5x + 4 = 0\):

   • This factors to \((x + 4)(x + 1) = 0\).
   • It has two distinct solutions: \(x = -4\) and \(x = -1\).

3. \(x^2 - 5x + 4 = 0\):

   • This factors to \((x - 4)(x - 1) = 0\).
   • It has two distinct solutions: \(x = 4\) and \(x = 1\).

4. \(x^2 - 4 = 0\):

   • This factors to \((x - 2)(x + 2) = 0\).
   • It has two distinct solutions: \(x = 2\) and \(x = -2\).

Therefore, the equation that has only one distinct solution is:

\(x^2 + 4x + 4 = 0\).