Find values of x for which 2x^2 < 5x + 12
Start by rearranging the inequality - make sure the sign next to the x2 term is positive to make it easier:
2x2 -5x - 12< 0
Next step is to factorise this quadratic. To do this, remember that the general form of a quadratic is ax2 + bx + c (in this case a =2, b = -5 and c = -12). Now, work out ac. This is 2x(-12) = -24, so we must find two factors of -24 that add together to give -5 (which is the coefficient of x). Notice that 24 = 1x24 = 2x12 = 3x8 = 4x6. We can see that to make -24, we can take our factors to be -8 and +3: add up to give -5.
Rewriting the quadratic: 2x2-5x-12 = 2x2 -8x +3x - 12 = 2x(x - 4) + 3(x -4). Here we have split -5x up into -8x and +3x, and then taken out common factors of 2x and 3 in the separate terms. We can now factorise out (x-4):
2x2-5x -12 = (x-4)(2x + 3) and hence the quadratic is factorised.
Next, let y = 2x2 -5x -12 and draw a graph: we can see that it's a U - shaped quadratic, and y = 0 when x = -3/2 and when x = 4. We want the values of x where y < 0: so by looking at the graph we want the region between these two roots: our answer is -3/2 < x < 4.
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