Unit 2Graphs and equationsStrategy:To factorise a quadratic of the formax2+bx+c1.Take out any numerical common factors. If the coeﬃcient ofx2isnegative, also take out the factor−1. Then apply the steps belowto the quadratic inside the brackets.2.Find the positive factor pairs ofa, the coeﬃcient ofx2. For eachsuch factor paird, ewrite down a framework (dx)(ex).3.Find all the factor pairs ofc, the constant term (including bothpositive and negative factors).4.For each framework and each factor pair ofc, write the factorpair in the gaps in the framework in both possible ways.5.For each of the resulting cases, calculate the term inxthat youobtain when you multiply out the brackets.6.Identify the case where this term isbx, if there is such a case.This is the required factorisation.As with the earlier strategy, if this strategy doesn’t lead to a factorisation,then the quadratic can’t be factorised using integers.Activity 18Factorising quadratics of the formax2+bx+cFactorise the following quadratics. (Theycanall be factorised.)(a) 5x2+ 13x−6(b) 3x2+ 16x+ 5(c) 6x2−11x+ 3(d) 5x2−8x−21(e) 18x2+ 9x−2(f) 4x2−8x+ 3(g) 4p2−19p−5(h) 6u2+ 11u−35(i) 4t2+ 4t+ 1(j) 9v2−12v+ 4(k)−4s2+ 4s+ 3(l) 12y2−10y−2There are two special types of quadratic that can be factorised more easilythan those that you have seen so far in this subsection. You should alwayscheck whether your quadratic is one of these before you embark on eitherof the two strategies above.Quadratics with no constant termThese can be factorised by takingxout as a common factor. For example,x2+ 4x=x(x+ 4),3x2−6x= 3(x2−2x) = 3x(x−2).162