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Solving Quadratics by Factorising

≈ 30 min
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The complete method, assembled from everything in this section:

  1. Rearrange to standard form: everything on one side, zero on the other.
  2. Factorise the quadratic (checklist: common factor → squares → trinomial).
  3. Zero-product: set each factor to zero.
  4. Solve the two mini-equations and check both roots.

Worked example. Solve x2−5x+6=0x^2 - 5x + 6 = 0.

Product 6, sum −5-5 → the pair −2,−3-2, -3:

(x−2)(x−3)=0  ⇒  x=2 or x=3(x - 2)(x - 3) = 0 \;\Rightarrow\; x = 2 \text{ or } x = 3

Worked example (rearrange first). Solve x2+3x=10x^2 + 3x = 10.

x2+3x−10=0  ⇒  (x+5)(x−2)=0  ⇒  x=−5 or x=2x^2 + 3x - 10 = 0 \;\Rightarrow\; (x + 5)(x - 2) = 0 \;\Rightarrow\; x = -5 \text{ or } x = 2

Solve by factorising: x2−5x+6=0x^2 - 5x + 6 = 0. Answer in the form x=… or x=…

Solve by factorising: x2+3x−10=0x^2 + 3x - 10 = 0. Answer in the form x=… or x=…

Find the positive root of x2−16=0x^2 - 16 = 0.

To solve x2+3x=10x^2 + 3x = 10, the correct FIRST move is…