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

Question

Responses

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

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

x2−4=0
x squared minus 4 equals 0

x2+5x+4=0

Answer 1

A quadratic equation has only one distinct solution when the discriminant is zero or when it factors into a perfect square trinomial.

Let's analyze each of the given equations:

1. \(x^2 - 5x + 4 = 0\)
   Factors to \((x - 1)(x - 4) = 0\)
   Solutions: \(x = 1\) and \(x = 4\) (two distinct solutions)

2. \(x^2 + 4x + 4 = 0\)
   Factors to \((x + 2)(x + 2) = 0\)
   Solution: \(x = -2\) (one distinct solution)

3. \(x^2 - 4 = 0\)
   Factors to \((x - 2)(x + 2) = 0\)
   Solutions: \(x = 2\) and \(x = -2\) (two distinct solutions)

4. \(x^2 + 5x + 4 = 0\)
   Factors to \((x + 1)(x + 4) = 0\)
   Solutions: \(x = -1\) and \(x = -4\) (two distinct solutions)

From this analysis, the equation that has only one distinct solution is:

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