Lesson WHY Factorise Quadratics?
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<b>Introduction</b> The aim of looking at an equation is to "solve" it. This means "put the equation equal to zero and find the value(s) if any of x that make the equation equal to zero. Now, physically, this means <b>"finding where the equation crosses the x-axis"</b>, since y=0 is the x-axis. That is all we do when a question says "solve". <b>Linear Example</b> Solve y = 2x - 6 This is asking... where does the straight line cross the x-axis... ie where is y=0? There are a couple ways of doing this: 1. Drawing a graph which is slow and not very accurate, in the majority of cases. In simple equations, it is a good enough method though... {{{ graph(300, 300, -4, 4, -10, 10, 2x - 6) }}} Looking at the graph, the answer is x=3 2. Algebraically 2x - 6 = 0 2x = 6 x = 6/2 x = 3 the benefit of this second method is it is quick and it shows you ALL the answers. Ok then, quadratics... <b>Quadratic Factorisation</b> All this build up for factorisation. But why? Well how about solving {{{ y=x^2 - 3x + 2 }}}? This again means "where does it cross the x-axis? ie where is y=0? And again, we can draw a graph, but this will potentially be imprecise, since we have a curve to draw now... {{{graph(300,300,-2,5,-5,5, x^2 - 3x + 2) }}} What are the 2 solutions to this, by looking at the graph? The roots, or solutions, seem to be at x=1 and x=2, but can we be sure by just looking at a graph? Now, algebraically, <b>IF</b> we could convert {{{ y=x^2 - 3x + 2 }}} into factors, ie factorise it, then we would have something useful... we would have: <b>FACTOR_1*FACTOR_2 = 0</b> In other words, two numbers multiply to equal zero. What are the two numbers that multiply to give zero? Well, to multiply and give zero as the answer, one of those two <b>HAS TO BE ZERO, ITSELF</b>. eg 2*4 = 8 12*-3 = -36 but 0*4 = 0 12*0 = 0 etc So, if we factorise {{{ x^2 - 3x + 2 }}} to (x-1)(x-2) and put it equal to zero, then we can say the following: <b>EITHER (x-1) = 0 OR (x-2) = 0</b> In either case, x=1 OR x=2. And in 3 lines of algebra, we have found all the EXACT roots (or solutions) of this quadratic. <b>Summary</b> This is where factorisation comes into its own: it is used to solve quadratic equations. As to how do we factorise quadratics, that is another Lesson, covered by others already.
