Question

Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point)
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+5x+4=0
x squared plus 5 x plus 4 equals 0

x2−4=0

Answer 1

To determine which of the given quadratic equations has only one distinct solution, we can look for the equation that can be factored into a perfect square. A quadratic equation has only one distinct solution when the discriminant is zero, or when it can be expressed as \((x - a)^2 = 0\).

Let's evaluate each option:

1. **\(x^2 - 5x + 4 = 0\)**
This factors to \((x - 4)(x - 1) = 0\).
Solutions: \(x = 4\) and \(x = 1\). (Two distinct solutions)

2. **\(x^2 + 4x + 4 = 0\)**
This factors to \((x + 2)(x + 2) = (x + 2)^2 = 0\).
Solution: \(x = -2\) (One distinct solution)

3. **\(x^2 + 5x + 4 = 0\)**
This factors to \((x + 4)(x + 1) = 0\).
Solutions: \(x = -4\) and \(x = -1\). (Two distinct solutions)

4. **\(x^2 - 4 = 0\)**
This can be factored as \((x - 2)(x + 2) = 0\).
Solutions: \(x = 2\) and \(x = -2\). (Two distinct solutions)

From the analysis, the equation with only one distinct solution is:

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