Question

What are the steps to factor a quadratic equation?

Answer 1

Factoring a quadratic equation typically involves rewriting it in the form \( ax^2 + bx + c \) as a product of two binomials. Here are the steps you can follow to factor a quadratic equation:

1. **Identify the quadratic equation**: The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

2. **Check for a common factor**: Before proceeding, see if there is a common factor among the coefficients \( a \), \( b \), and \( c \). If there is, factor it out first.

3. **Calculate the product \( ac \)**: Multiply \( a \) (the coefficient of \( x^2 \)) by \( c \) (the constant term). This product will help in the next step.

4. **Find two numbers that multiply to \( ac \) and add to \( b \)**: Look for two numbers that multiply to \( ac \) and add up to \( b \) (the coefficient of \( x \)). Let’s call these numbers \( m \) and \( n \).

5. **Rewrite the middle term**: Rewrite the quadratic by splitting the middle term \( bx \) into two terms using \( m \) and \( n \). The equation will now look like this:
\[
ax^2 + mx + nx + c
\]

6. **Factor by grouping**: Group the terms into two pairs:
\[
(ax^2 + mx) + (nx + c)
\]
Factor out the common factors from each group:
\[
x(a + m) + k(n + c)
\]

7. **Factor out the common binomial**: If done correctly, you will have a common binomial factor. Factor it out:
\[
(px + q)(rx + s)
\]

8. **Check your work**: Finally, you can verify your factorization by expanding \((px + q)(rx + s)\) to ensure that you obtain the original quadratic equation.

**Example**:
Let's factor the quadratic equation \( x^2 + 5x + 6 \):

1. Here, \( a = 1\), \( b = 5\), \( c = 6\).
2. There is no common factor.
3. Calculate \( ac = 1 \cdot 6 = 6 \).
4. Find two numbers that multiply to \( 6 \) and add to \( 5 \): Those numbers are \( 2 \) and \( 3 \).
5. Rewrite the equation: \( x^2 + 2x + 3x + 6 \).
6. Group the terms: \( (x^2 + 2x) + (3x + 6) \).
7. Factor each group: \( x(x + 2) + 3(x + 2) \).
8. Factor out the common binomial: \( (x + 2)(x + 3) \).

Thus, the factorization of \( x^2 + 5x + 6 \) is \( (x + 2)(x + 3) \).