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

Question

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 quadratic equation has only one distinct solution, we need to understand that this happens when the equation can be factored into a perfect square. A quadratic equation of the form \( ax^2 + bx + c = 0 \) has only one distinct solution when the discriminant (\( b^2 - 4ac \)) is equal to zero.

Let's analyze each equation:

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

   • Factoring: \( (x - 2)(x + 2) = 0 \)
   • Solutions: \( x = 2, x = -2 \) (two distinct solutions)

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

   • Factoring: \( (x - 4)(x - 1) = 0 \)
   • Solutions: \( x = 4, x = 1 \) (two distinct solutions)

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

   • Factoring: \( (x + 4)(x + 1) = 0 \)
   • Solutions: \( x = -4, x = -1 \) (two distinct solutions)

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

   • Factoring: \( (x + 2)(x + 2) = 0 \) or \( (x + 2)^2 = 0 \)
   • Solution: \( x = -2 \) (one distinct solution)

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

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