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

Question

(1 point)
Responses

x2 − 5x + 4 = 0
x2 − 5x + 4 = 0

x2 + 4x + 4 = 0
x2 + 4x + 4 = 0

x2 + 5x + 4 = 0
x2 + 5x + 4 = 0

x2 − 4 = 0

Answer 1

A quadratic equation has only one distinct solution when its discriminant is zero. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

Let's analyze each of the equations to find the one with a discriminant of zero.

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

   • \( a = 1, b = -5, c = 4 \)
   • Discriminant: \( D = (-5)^2 - 4(1)(4) = 25 - 16 = 9 \) (two distinct solutions)

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

   • \( a = 1, b = 4, c = 4 \)
   • Discriminant: \( D = (4)^2 - 4(1)(4) = 16 - 16 = 0 \) (one distinct solution)

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

   • \( a = 1, b = 5, c = 4 \)
   • Discriminant: \( D = (5)^2 - 4(1)(4) = 25 - 16 = 9 \) (two distinct solutions)

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

   • This can be rewritten as \( (x - 2)(x + 2) = 0 \)
   • Discriminant: \( D = 0 \) (two distinct solutions, \( x = 2 \) and \( x = -2 \))

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

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