Section 7.5: Factoring Quadratics
Quadratic trinomials can often be factored into a product of two binomials. Start by identifying a, b, c in \( ax^2 + bx + c \) and find two numbers that multiply to \( a \cdot c \) and sum to b.
Example 1
Factor: \( x^2 + 5x + 6 \)
Find two numbers multiplying to 6 (c) and adding to 5 (b): 2 and 3
Factor: \( (x + 2)(x + 3) \)
Example 2
Factor: \( 2x^2 + 7x + 3 \)
Multiply a * c: 2 * 3 = 6. Find numbers adding to 7: 6 and 1
Split middle term: \( 2x^2 + 6x + x + 3 \)
Factor by grouping: \( 2x(x + 3) + 1(x + 3) \)
Factor out common binomial: \( (2x + 1)(x + 3) \)
Practice Problems
- Factor: \( x^2 + 8x + 15 \)
- Factor: \( 3x^2 + 11x + 6 \)
- Factor: \( 4x^2 - 12x + 9 \)
- Factor: \( 5x^2 + 9x + 4 \)
- Factor: \( x^2 - 7x + 12 \)