To analyze the quadratic equation \( x^2 + 6x + 9 = 0 \), first, we need to calculate the discriminant. The discriminant (\( D \)) of a quadratic equation in the form \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
For the equation \( x^2 + 6x + 9 = 0 \):
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)
Substituting these values into the discriminant formula:
\[ D = 6^2 - 4 \cdot 1 \cdot 9 \] \[ D = 36 - 36 \] \[ D = 0 \]
Since the discriminant is \( 0 \), this indicates that the quadratic equation has exactly one real solution (or one repeated real root).
Therefore, we can conclude:
- This quadratic equation will have one real solution because the discriminant is zero.
So the phrases that are true about the solutions to this quadratic equation are:
- "This quadratic equation will have one real solution"
- "because the discriminant is zero"