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

x2−4=0
x squared minus 4 equals 0

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

x2+5x+4=0

Answer 1

To determine which quadratic equation has only one distinct solution, we can look for a situation where the quadratic can be factored into a perfect square. This occurs when the discriminant (b² - 4ac) is zero.

Let's analyze each equation:

1. x² - 5x + 4 = 0
   Factors: (x - 1)(x - 4) = 0
   Solutions: x = 1, x = 4 (two distinct solutions)

2. x² - 4 = 0
   Factors: (x - 2)(x + 2) = 0
   Solutions: x = 2, x = -2 (two distinct solutions)

3. x² + 4x + 4 = 0
   Factors: (x + 2)(x + 2) = 0 (or (x + 2)² = 0)
   Solutions: x = -2 (one distinct solution)

4. x² + 5x + 4 = 0
   Factors: (x + 1)(x + 4) = 0
   Solutions: x = -1, x = -4 (two distinct solutions)

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

x² + 4x + 4 = 0 (which factors to (x + 2)² = 0, yielding the double root x = -2).