Advanced algebra

7.1 · Free notes

Factorising quadratics and the difference of two squares

Key points

• A quadratic expression has the form ax² + bx + c (a ≠ 0). Factorising writes it as a product of brackets, which is the reverse of expanding.
• To factorise x² + bx + c, find two numbers that multiply to c and add to b. For example x² + 7x + 12 = (x + 3)(x + 4), since 3 × 4 = 12 and 3 + 4 = 7.
• Check the signs. If c is positive, both numbers have the same sign as b. If c is negative, the numbers have different signs, and the larger one takes the sign of b.
• Always take out a common factor first. 3x² + 6x = 3x(x + 2) and 2x² − 18 = 2(x − 3)(x + 3). "Factorise fully" means no further factors remain.
• The difference of two squares is a² − b² = (a − b)(a + b). For example x² − 49 = (x − 7)(x + 7). It works only for one square minus another square.
• x² + 49 (a sum of squares) cannot be factorised using real numbers. There is no middle term, so DOTS does not apply.
• With coefficients, 4x² − 25 = (2x − 5)(2x + 5) and 9x² − 16y² = (3x − 4y)(3x + 4y). Take the square root of each term, including its coefficient.
• For ax² + bx + c with a ≠ 1, multiply a × c and find two numbers that multiply to ac and add to b. Split the middle term and factorise in pairs: 2x² + 7x + 3 = 2x² + 6x + x + 3 = (2x + 1)(x + 3).
• Check by expanding your brackets (for example with FOIL or a grid). You must get the original expression back.
• To solve a quadratic equation by factorising, first rearrange to = 0. Factorise, then set each bracket equal to 0. (x − 3)(x + 4) = 0 gives x = 3 or x = −4. Note the sign change.
• Use DOTS for quick arithmetic: 101² − 99² = (101 − 99)(101 + 99) = 2 × 200 = 400. You can also use factorising to simplify algebraic fractions, for example (x² − 9)/(x² + x − 6) = (x − 3)/(x − 2).

In more detail

• Worked simple quadratic: x² + 2x − 15 = (x + 5)(x − 3), because 5 × (−3) = −15 and 5 + (−3) = 2.
• Worked a ≠ 1: 6x² + 7x − 3: ac = −18, use 9 and −2 → 6x² + 9x − 2x − 3 = 3x(2x + 3) − 1(2x + 3) = (3x − 1)(2x + 3).
• Worked DOTS: 9x² − 16y² = (3x − 4y)(3x + 4y); 2x² − 50 = 2(x² − 25) = 2(x − 5)(x + 5).
• Worked mental arithmetic: 53² − 47² = (53 − 47)(53 + 47) = 6 × 100 = 600.
• Worked solving: 3x² − 12 = 0 → 3(x − 2)(x + 2) = 0 → x = 2 or x = −2.