HIGHER TIER a) Factorise the following equation into two bracket form: 2x^2-5x-12. b)2x^2-5x-12=0. Solve this equation to find the values of x, using your answer to part a). BONUS c) Sketch the function y=2x^2-5x-12, showing any x intercepts

a) Find two common numbers that ADD UP to give -5, and MULTIPLY TOGETHER to give (2*-12=) -24MULTIPLY TO -24 -1 and 24 1 and -24 -2 and 12 2 and -12 3 and -8 -3 and 8 4 and -6 - 4 and 6ADD TO -5 5 and -10 4 and -9 3 and -8 2 and -7 1 and -6- 1 and -4 -2 and -3Then split the original equation according to this pair: 2x²-5x-12 = 2x²-8x+3x-12 Then, factorise each half: 2x²-8x+3x-12 = 2x(x-4) + 3(x-4) Finally, split this up into two brackets to give the answer: (2x+3)(x-4)b) Substituting in the answers from above, (2x+3)(x-4)=0 Therefore 2x = -3 and x=4 Therefore x=-3/2 and x=4c) BONUSThe key features to the sketch are: Present in all four quadrants, ‘U’ shape, crosses x axis at the answers from part b) Ie at x= -3/2, and x=4EXTRA POINT - substitute in x=0 to find y axis intercept I.e y= -12EXTRA POINT - find the coordinates of the minimum of this graph. As the arithmetic in this question is challenging, it is not expected that students use completing the square to work out the answer to this. Instead, symmetry should be used:The x coordinate of the minimum will be halfway between the two answers found in d) - i.e at 5/4 (1.25)The y coordinate can then be found by plugging in 5/4 into the original equation to give y = -15.125.The arithmetic here is very challenging! This would be a worthwhile task for a student to push themselves in preparation for exams, but is unlikely to come up in such difficult in any normal mathematics GCSE examination.