Question

how do you factor a quadratic equations

Answer 1

Here are clear steps and examples for factoring a quadratic expression ax^2 + bx + c.

General steps
1. Factor out any greatest common factor (GCF).
2. If a = 1 (so you have x^2 + bx + c): find two integers m and n with m·n = c and m + n = b. Then x^2 + bx + c = (x + m)(x + n).
3. If a ≠ 1: use the "ac method" (split-the-middle) or trial-and-error. For ac method:
a. Compute ac = a·c.
b. Find two integers p and q with p·q = ac and p + q = b.
c. Rewrite bx as px + qx, then factor by grouping.
4. Check by expanding (FOIL) that your factors multiply back to ax^2 + bx + c.
5. If you cannot find integer p and q, the polynomial may not factor over the integers; use the quadratic formula x = [-b ± sqrt(b^2 - 4ac)]/(2a) to find real or complex roots and then write factors as a(x − r1)(x − r2) if desired.

Examples
- a = 1: x^2 + 5x + 6. Find m, n with m·n = 6 and m + n = 5 → 2 and 3. Factor: (x + 2)(x + 3).

- a ≠ 1 (ac method): 6x^2 + 11x + 3. ac = 18. Find p, q with p·q = 18 and p + q = 11 → 2 and 9. Rewrite:
6x^2 + 2x + 9x + 3 = 2x(3x + 1) + 3(3x + 1) = (2x + 3)(3x + 1).

- GCF: 2x^2 + 4x = 2x(x + 2).

- Special patterns:
- Perfect square: x^2 + 6x + 9 = (x + 3)^2.
- Difference of squares: x^2 − 9 = (x − 3)(x + 3).

When it doesn't factor nicely
- Example: x^2 + 4x + 7. No integer pair m, n multiply to 7 and add to 4 → use quadratic formula to find complex or irrational roots.

If you want, tell me a specific quadratic and I’ll factor it step by step.