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Factoring quadratic equations is often the most challenging part of algebra. It requires a solid grasp of algebraic terminology and multi‑step linear equations. There are three main techniques—factoring, graphing, and the quadratic formula—and the questions you ask differ depending on the method.
Before you begin, confirm that the equation is in the standard form ax² + bx + c = 0, with a ≠ 0. If the right‑hand side contains terms, move them to the left side. For example, from 3x² – x – 4 = 6, subtract 6 to get 3x² – x – 10 = 0.
When a = 1, factoring is often the quickest. If a ≠ 1, consider another method first. To factor, find two numbers that multiply to c and add to b. For instance, (x – 9)(x + 4) = 0 solves x² – 5x – 36 = 0 because –9 × 4 = –36 and –9 + 4 = –5.
Graphing is useful if you have a graphing calculator. After entering the equation, ensure the window includes the x‑intercepts. For x² – 11x – 26 = 0, the graph shows one root at x = –2. Adjust the window to see the second root at x = 13.
The quadratic formula works for every quadratic, including irrational or complex roots:
x = [–b ± √(b² – 4ac)] ÷ (2a)
Insert the correct a, b, c values and watch the sign of b. For 8x² – 22x – 6 = 0, a = 8, b = –22, c = –6. The formula becomes x = [22 ± √(484 – 4(8)(–6))] ÷ 16, yielding x = 3 and x = –0.25.
See Reference 1 or Reference 2.
