Question

How do I use the quadratic formula to solve the equation x^2 + 5x + 6 = 0?

Answer 1

To solve the quadratic equation \(x^2 + 5x + 6 = 0\) using the quadratic formula, you can follow these steps:

1. **Identify the coefficients**: The quadratic equation is in the standard form \(ax^2 + bx + c = 0\). For your equation:
- \(a = 1\)
- \(b = 5\)
- \(c = 6\)

2. **Write the quadratic formula**: The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\]

3. **Calculate the discriminant** (\(b^2 - 4ac\)):
\[
b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1
\]

4. **Substitute the values into the quadratic formula**:
- Since the discriminant is positive (\(1\)), there will be two real and distinct solutions.
\[
x = \frac{{-5 \pm \sqrt{1}}}{2 \cdot 1}
\]

5. **Calculate the two possible values for \(x\)**:
\[
x = \frac{{-5 + 1}}{2} = \frac{{-4}}{2} = -2
\]
\[
x = \frac{{-5 - 1}}{2} = \frac{{-6}}{2} = -3
\]

6. **Final solutions**: The solutions to the equation \(x^2 + 5x + 6 = 0\) are:
\[
x = -2 \quad \text{and} \quad x = -3
\]

These are the roots of the quadratic equation.