Question

Consider the quadratic equation below.



x
2
+
6
x
+
9
=
0
x
2
+6x+9=0



Select the phrases that are true about the solutions to this quadratic equation.



This quadratic equation will have
because the discriminant is

Answer 1

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"