الأسبوع 5 · التحليل: العامل المشترك الأكبر ثم ثلاثية الحدود والمعادلات التربيعية · 3 دروس
3
Factorising ax² + bx + c and solving quadratic equations
Mathematics Prep 2 Term 1, Unit 2, Lessons 1–2
GoalFactorise ax² + bx + c by trial, take out a common factor first, and solve quadratic equations by factorising.
New words
quadratic equation · an equation whose highest power is x², such as x² + 13x + 30 = 0trial method · try the pairs of factors and test the middle termroots · the values of x that make the equation truemiddle term · the term bx in the middle
Explanation
When the first coefficient is not 1, take out any common factor first: 3x² + 6x − 45 = 3(x² + 2x − 15) = 3(x + 5)(x − 3). Otherwise try the possible pairs of brackets. For 2x² + 7x + 6 the first terms are 2x and x, and the last terms multiply to 6. Test the middle term (outer product plus inner product): (2x + 1)(x + 6) gives 13x, (2x + 6)(x + 1) gives 8x, (2x + 2)(x + 3) gives 8x, and (2x + 3)(x + 2) gives 4x + 3x = 7x. So 2x² + 7x + 6 = (2x + 3)(x + 2). To solve a quadratic equation, move all terms to one side so the other side is 0, factorise, and use the zero-product rule. x² + 13x + 30 = 0 gives (x + 3)(x + 10) = 0, so x = −3 or x = −10. Substitute to check.
Try
Middle term
(2x + 1)(x + 6)
13x
(2x + 6)(x + 1)
8x
(2x + 2)(x + 3)
8x
(2x + 3)(x + 2)
7x
Worked examples
Factorise 6x² − 13x + 6. Both signs are negative. Try (2x − 3)(3x − 2): the middle term is −4x − 9x = −13x. It works: 6x² − 13x + 6 = (2x − 3)(3x − 2). So 6x² − 13x + 6 = 0 gives x = 3/2 or x = 2/3.
Solve x² + 13x + 30 = 0. The numbers 3 and 10 have product 30 and sum 13, so (x + 3)(x + 10) = 0 and x = −3 or x = −10. Check x = −3: 9 − 39 + 30 = 0.
A rectangle has area 48 m² and its length is 2 m more than its width w. Then w(w + 2) = 48, so w² + 2w − 48 = 0 and (w + 8)(w − 6) = 0. Width cannot be −8, so w = 6 m and the length is 8 m. Check: 6 × 8 = 48.
Class activity
Rectangle riddles. A rectangle has width w and length w + 3. Make a table of w = 1 to 8 with the area w(w + 3). For which w is the area 40? Solve w² + 3w − 40 = 0 by factorising and compare.
