You're correct! The solutions to the quadratic equation \( x^2 + 5x + 6 = 0 \) can be found by factoring the equation.
The quadratic can be factored as:
\[ (x + 2)(x + 3) = 0 \]
Setting each factor equal to zero gives us the solutions:
- \( x + 2 = 0 \) leads to \( x = -2 \)
- \( x + 3 = 0 \) leads to \( x = -3 \)
Thus, the solutions are indeed \( x = -3 \) and \( x = -2 \).
You can also verify the solutions by using the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
For this specific equation:
- \( a = 1 \)
- \( b = 5 \)
- \( c = 6 \)
Plugging in these values:
\[ x = \frac{{-5 \pm \sqrt{{5^2 - 4 \cdot 1 \cdot 6}}}}{2 \cdot 1} \] \[ x = \frac{{-5 \pm \sqrt{{25 - 24}}}}{2} \] \[ x = \frac{{-5 \pm 1}}{2} \]
This gives the two solutions:
- \( x = \frac{{-5 + 1}}{2} = \frac{{-4}}{2} = -2 \)
- \( x = \frac{{-5 - 1}}{2} = \frac{{-6}}{2} = -3 \)
So, both methods confirm that the solutions are \( x = -3 \) and \( x = -2 \).