Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point) Responses x2−4=0 x squared minus 4 equals 0 x2−5x+4=0 x squared minus 5 x plus 4 equals 0 x2+5x+4=0 x squared plus 5 x plus 4 equals 0 x2+4x+4=0

Answer 1

To determine which of the given quadratic equations has only one distinct solution, we look for an equation that can be factored into a perfect square. A quadratic equation has one distinct solution when its discriminant (\(b^2 - 4ac\)) is zero.

Let's analyze each equation:

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

   • Factors as: \((x - 2)(x + 2) = 0\)
   • Solutions: \(x = 2\) and \(x = -2\) (two distinct solutions)

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

   • Factors as: \((x - 4)(x - 1) = 0\)
   • Solutions: \(x = 4\) and \(x = 1\) (two distinct solutions)

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

   • Factors as: \((x + 4)(x + 1) = 0\)
   • Solutions: \(x = -4\) and \(x = -1\) (two distinct solutions)

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

   • Factors as: \((x + 2)(x + 2) = 0\) (this is a perfect square)
   • Solutions: \(x = -2\) (only one distinct solution)

Since \(x^2 + 4x + 4 = 0\) factors into a perfect square, it has only one distinct solution.

Therefore, the equation with only one distinct solution is:

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