Overview: Factoring ax² + bx + c
Factoring trinomials of the form ax² + bx + c where a ≠ 1 requires either the trial-and-error method or the AC method, both covered in enVision Algebra 1 Chapter 7 for Grade 11. Trial and error tests factor pairs of a and c until the correct combination for the middle term is found: 2x² + 7x + 3 factors as (2x+1)(x+3). The AC method finds two numbers that multiply to ac and add to b, then rewrites the middle term: for 6x² + 7x - 5, find factors of -30 that sum to 7 (10 and -3), rewrite as 6x² + 10x - 3x - 5, then factor by grouping to get (2x-1)(3x+5).
Key Concepts
Trinomials of the form $ax^2 + bx + c$ where $a \neq 1$ : Use the trial and error method or the 'ac' method to find factors. Trial and Error Method : Test different factor combinations of the form $(px + m)(qx + n)$ where $pq = a$ and $mn = c$. AC Method : Find two numbers that multiply to $ac$ and add to $b$, then rewrite the middle term and factor by grouping.
Common Questions
How does the trial and error method work for factoring ax² + bx + c?
Test pairs (px + m)(qx + n) where pq = a and mn = c. Check if the outer and inner products combine to give bx. For 2x² + 7x + 3, (2x+1)(x+3) works: outer 6x + inner x = 7x.
What are the steps of the AC method for 6x² + 7x - 5?
Multiply a·c = 6·(-5) = -30. Find two numbers that multiply to -30 and add to 7: those are 10 and -3. Rewrite as 6x² + 10x - 3x - 5, factor by grouping: 2x(3x+5) - 1(3x+5) = (2x-1)(3x+5).
How do you factor 2x² + 7x + 3 by trial and error?
Factor pairs of 2 (the a coefficient): just (2,1). Factor pairs of 3: (1,3) or (3,1). Test (2x+1)(x+3) = 2x² + 7x + 3. This works.
How do you verify a factorization is correct?
Expand the factors using FOIL or the distributive property and confirm you get the original trinomial back.
When should you use the AC method vs. trial and error?
Use trial and error when a and c have few factor pairs, making guessing quick. Use the AC method when there are many possible combinations to test, as it is more systematic and always finds the answer.