Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point)

Responses

x2+5x+4=0
x squared plus 5 x plus 4 equals 0

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

x2−4=0
x squared minus 4 equals 0

x2−5x+4=0

1 answer

A quadratic equation has only one distinct solution when it can be factored into a perfect square. This occurs when the discriminant (\(b^2 - 4ac\)) is equal to zero.

Let's analyze each of the given equations:

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

    • Discriminant: \(5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9\) (two distinct solutions)
  2. \(x^2 + 4x + 4 = 0\)

    • Discriminant: \(4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0\) (one distinct solution)
  3. \(x^2 - 4 = 0\)

    • Discriminant: \(0^2 - 4 \cdot 1 \cdot (-4) = 0 + 16 = 16\) (two distinct solutions)
  4. \(x^2 - 5x + 4 = 0\)

    • Discriminant: \((-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9\) (two distinct solutions)

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

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