Overview
This lecture explains five methods to factor quadratic trinomials with a leading coefficient not equal to one, using the example 6x² + 7x - 3.
AC Method With Grouping
- Multiply the leading coefficient (a) and constant term (c): 6 * -3 = -18.
- Find two numbers that multiply to -18 and add to 7; these are 9 and -2.
- Rewrite 7x as -2x + 9x to get four terms: 6x² - 2x + 9x - 3.
- Group and factor: (6x² - 2x) + (9x - 3).
- Factor out GCFs: 2x(3x - 1) + 3(3x - 1).
- Factor common binomial: (3x - 1)(2x + 3).*
AC Method With Box
- Still use a * c = -18; numbers are -2 and 9.
- Draw a 2x2 box: fill with 6x², -2x, 9x, -3.
- Factor rows and columns to get factors: (3x - 1)(2x + 3).*
Parenthesis Reduction Method
- Start with parentheses: (6x - 2)(6x + 9).
- Reduce each factor to lowest terms: 6x - 2 becomes 3x - 1, 6x + 9 becomes 2x + 3.
- Final factors: (3x - 1)(2x + 3).
Slide and Divide Method (Amazon Prime Way)
- Multiply a * c and create new quadratic: x² + 7x - 18.
- Factor into (x - 2)(x + 9).
- Divide each number by original a (6): get (x - 2/6)(x + 9/6).
- Reduce fractions, rewrite as binomials: (3x - 1)(2x + 3).*
Tic-Tac-Toe Method
- Draw a 2x2 grid and fill with possible combinations to multiply to a and c.
- Check sums to match middle term (7x).
- Correct combination results in (2x + 3)(3x - 1).
Key Terms & Definitions
- Quadratic Trinomial — a polynomial of the form ax² + bx + c.
- Leading Coefficient (a) — the coefficient before x² in a quadratic.
- AC Method — factoring by multiplying a and c and finding two numbers that multiply to ac and sum to b.
- Grouping — breaking up the middle term and factoring in pairs.
- Tic-Tac-Toe Method — a box/grid method to organize factor pairs.
Action Items / Next Steps
- Practice factoring more quadratic trinomials using each method.
- Choose the method you find most intuitive for future problems.