Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point) Responses x^2+5x+4=0 x squared plus 5 x plus 4 equals 0 x^2−4=0 x squared minus 4 equals 0 x^2+4x+4=0 x squared plus 4 x plus 4 equals 0 x^2−5x+4=0

Answer 1

A quadratic equation has only one distinct solution when it can be expressed as a perfect square trinomial, meaning it can be factored into a form like \((x - r)^2 = 0\).

Let's analyze each equation:

1. \(x^2 + 5x + 4 = 0\): Factors to \((x + 4)(x + 1) = 0\) (two distinct solutions).
2. \(x^2 - 4 = 0\): Factors to \((x - 2)(x + 2) = 0\) (two distinct solutions).
3. \(x^2 + 4x + 4 = 0\): This factors to \((x + 2)^2 = 0\) (one distinct solution, \(x = -2\)).
4. \(x^2 - 5x + 4 = 0\): Factors to \((x - 4)(x - 1) = 0\) (two distinct solutions).

The equation that has only one distinct solution is:

\(x^2 + 4x + 4 = 0\), which gives the solution \(x = -2\).