Question

How to solve the quadratic equation x^2 + 5x + 6 = 0?

Answer 1

To solve the quadratic equation \( x^2 + 5x + 6 = 0 \), we can use the factoring method.

1. **Identify the coefficients**: In the equation \( ax^2 + bx + c = 0 \):
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 5 \) (coefficient of \( x \))
- \( c = 6 \) (constant term)

2. **Find two numbers that multiply to \( c \) (6) and add to \( b \) (5)**:
- The pair of numbers that satisfy these conditions are \( 2 \) and \( 3 \) because:
- \( 2 \times 3 = 6 \)
- \( 2 + 3 = 5 \)

3. **Factor the quadratic**:
The equation can be factored as:
\[
(x + 2)(x + 3) = 0
\]

4. **Set each factor to zero**:
Now, we can solve for \( x \) by setting each factor to zero:
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]

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

You can also verify the solutions by substituting them back into the original equation. Both \( x = -2 \) and \( x = -3 \) will satisfy \( x^2 + 5x + 6 = 0 \).