Using Quadratic Equations
12-5Methods of Solution
Overview
Three ways: factoring, completing the square, quadratic formula. Choose what’s easiest.
Key ideas
- Factor when factors are easy to spot.
- Use the formula when factoring is hard.
- Complete the square to derive the formula or rewrite in vertex form.
Worked examples
Example 1
Problem. Solve x² − x − 6 = 0.
Solution. Factor: (x − 3)(x + 2) = 0 → x = 3 or −2.
Example 2
Problem. Solve x² + 4x + 1 = 0.
Solution. Quadratic formula: x = −2 ± √3.
Practice
Try each one. Click Show answer when ready.
- 1.
Solve x² − 9 = 0.
- 2.
Solve x² − 2x − 8 = 0.
- 3.
Solve x² − 6x + 7 = 0.
Challenge problems
A little tougher — great for test prep. Click Show answer when ready.
- 1.
Solve x⁴ − 5x² + 4 = 0 by substituting u = x².
- 2.
Solve x² − 2x − 8 = 0 by any method.