Question

How do I solve the quadratic equation x^2 + 5x + 6 and find similar questions involving factoring quadratic expressions?

Answer 1

To solve the quadratic equation \( x^2 + 5x + 6 = 0 \) by factoring, you want to express the quadratic in the form \( (x + p)(x + q) = 0 \), where \( p \) and \( q \) are numbers that satisfy two conditions:

1. \( p + q = 5 \) (the coefficient of \( x \))
2. \( p \times q = 6 \) (the constant term)

Let's find two numbers that satisfy these conditions.

The pairs of factors of \( 6 \) are:
- \( 1 \) and \( 6 \) (sum: \( 1 + 6 = 7 \))
- \( 2 \) and \( 3 \) (sum: \( 2 + 3 = 5 \))

Since the pair \( (2, 3) \) meets both conditions, we can factor the quadratic as:
\[
(x + 2)(x + 3) = 0
\]

Now, to solve for \( x \), set each factor to zero:
1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( x + 3 = 0 \) → \( x = -3 \)

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

### Finding Similar Questions Involving Factoring Quadratic Expressions

Here are a few similar quadratic equations that can be solved by factoring:

1. **Equation:** \( x^2 + 7x + 10 = 0 \)
- Numbers that satisfy \( p + q = 7 \) and \( p \times q = 10 \) are \( 2 \) and \( 5 \).

2. **Equation:** \( x^2 + 8x + 15 = 0 \)
- Numbers that satisfy \( p + q = 8 \) and \( p \times q = 15 \) are \( 3 \) and \( 5 \).

3. **Equation:** \( x^2 + 4x - 5 = 0 \)
- Look for numbers that satisfy \( p + q = 4 \) and \( p \times q = -5 \); these will be \( 5 \) and \( -1 \).

4. **Equation:** \( x^2 - 3x - 10 = 0 \)
- Here, you're looking for numbers that satisfy \( p + q = -3 \) and \( p \times q = -10 \); these will be \( 2 \) and \( -5 \).

5. **Equation:** \( x^2 - 5x + 6 = 0 \)
- Numbers that satisfy \( p + q = -5 \) and \( p \times q = 6 \) are \( -2 \) and \( -3 \).

You can create more such problems by choosing different values for \( p \) and \( q \) and ensuring they give you a quadratic expression that you can factor.