Question

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

x^2+5x+4=0
x squared plus 5 x plus 4 equals 0

x^2−4=0
x squared minus 4 equals 0

x^2+4x+4=0
x squared plus 4 x plus 4 equals 0

x^2−5x+4=0

Answer 1

To determine which quadratic equation has only one distinct solution, we can analyze them by checking their discriminants. The discriminant \(D\) of a quadratic equation in the standard form \(ax^2 + bx + c = 0\) is given by the formula:

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

A quadratic equation has one distinct solution if the discriminant is equal to zero (\(D = 0\)).

Let's calculate the discriminant for each given equation:

1. **For \(x^2 + 5x + 4 = 0\)**:
- \(a = 1\), \(b = 5\), \(c = 4\)
- \(D = 5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9\) (two distinct solutions)

2. **For \(x^2 - 4 = 0\)**:
- \(a = 1\), \(b = 0\), \(c = -4\)
- \(D = 0^2 - 4 \cdot 1 \cdot (-4) = 0 + 16 = 16\) (two distinct solutions)

3. **For \(x^2 + 4x + 4 = 0\)**:
- \(a = 1\), \(b = 4\), \(c = 4\)
- \(D = 4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0\) (one distinct solution)

4. **For \(x^2 - 5x + 4 = 0\)**:
- \(a = 1\), \(b = -5\), \(c = 4\)
- \(D = (-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9\) (two distinct solutions)

Based on the calculations, **the equation with only one distinct solution is**:

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