Common Factor First check if there any common factors. Example: what are the factors of 6x2 β 2x = 0 ? 6 and 2 have a common factor of 2: 2(3x2 β x) = 0 And x2 and x have a common factor of x: 2x(3x β 1) = 0 And we have done it! The factors are 2x and 3x β 1, We can now also find the roots (where it equals zero): 2x will be 0 when x = 0 3x β 1 will be zero when x = 1/3 And this is the graph (see how it is zero at x=0 and x=1/3): But it is not always that easy ... Guess and Check Maybe we can guess an answer? Example: what are the factors of 2x2 + 7x + 3 ? No common factors. Let us try to guess an answer, and then check if we are right ... we might get lucky! We could guess (2x+3)(x+1): (2x+3)(x+1) = 2x2 + 2x + 3x + 3 = 2x2 + 5x + 3 (WRONG) How about (2x+7)(x-1): (2x+7)(x-1) = 2x2 - 2x + 7x - 7 = 2x2 + 5x - 7 (WRONG AGAIN) OK, how about (2x+9)(x-1): (2x+9)(x-1) = 2x2 - 2x + 9x - 9 = 2x2 + 7x - 9 (WRONG AGAIN) Oh No! We could be guessing for a long time before we get lucky. That is not a very good method. So let us try something else: A Method For Simple Cases Luckily there is a method that works in simple cases. With the quadratic equation in this form: Quadratic Equation Step 1: Find two numbers that multiply to give ac (in other words a times c), and add to give b. Example: 2x2 + 7x + 3 ac is 2Γ3 = 6 and b is 7 So we want two numbers that multiply together to make 6, and add up to 7 In fact 6 and 1 do that (6Γ1=6, and 6+1=7) How do we find 6 and 1? It helps to list the factors of ac=6, and then try adding some to get b=7. Factors of 6 include 1, 2, 3 and 6. Aha! 1 and 6 add to 7, and 6Γ1=6. Step 2: Rewrite the middle with those numbers: Rewrite 7x with 6x and 1x: 2x2 + 6x + x + 3 Step 3: Factor the first two and last two terms separately: The first two terms 2x2 + 6x factor into 2x(x+3) The last two terms x+3 dont actually change in this case So we get: 2x(x+3) + (x+3) Step 4: If weve done this correctly, our two new terms should have a clearly visible common factor. In this case we can see that (x+3) is common to both terms So we can now rewrite it like this: 2x(x+3) + (x+3) = (2x+1)(x+3) Check: (2x+1)(x+3) = 2x2 + 6x + x + 3 = 2x2 + 7x + 3 (Yes) Much better than guessing! Let us try another example: Example: 6x2 + 5x - 6 Step 1: ac is 6Γ(-6) = -36, and b is 5 List the positive factors of ac = -36: 1, 2, 3, 4, 6, 9, 12, 18, 36 One of the numbers has to be negative to make -36, so by playing with a few different numbers I find that -4 and 9 work nicely: -4Γ9 = -36 and -4+9 = 5 Step 2: Rewrite 5x with -4x and 9x: 6x2 - 4x + 9x - 6 Step 3: Factor first two and last two: 2x(3x - 2) + 3(3x -2) Step 4: Common Factor is (3x - 2): (2x+3)(3x - 2) Check: (2x+3)(3x - 2) = 6x2 - 4x + 9x - 6 = 6x2 + 5x - 6 (Yes) Finding Those Numbers The hardest part is finding two numbers that multiply to give ac, and add to give b. It is partly guesswork, and it helps to list out all the factors. Here is another example to assist you: Example: ac = -120 and b = 7 What two numbers multiply to -120 and add to 7 ? The factors of 120 are (plus and minus): 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120 We can try pairs of factors (start near the middle!) and see if they add to 7: -10 x 12 = -120, and -10+12 = 2 (no) -8 x 15 = -120 and -8+15 = 7 (YES!) Why Factor? Well, one of the big benefits of factoring is that we can find the roots of the quadratic equation (where the equation is zero). All we need to do (after factoring) is find where each of the two factors becomes zero Example: what are the roots (zeros) of 6x2 + 5x β 6 ? We already know (from above) the factors are (2x + 3)(3x β 2) And we can figure out that (2x + 3) would be zero when x = β3/2 and (3x β 2) would be zero when x = 2/3 So the roots of 6x2 + 5x β 6 are: β3/2 and 2/3 Here is a plot of 6x2 + 5x β 6, can you see where it equals zero? And we can also check it using a bit of arithmetic: At x = -3/2: 6(-3/2)2 + 5(-3/2) - 6 = 6Γ(9/4) - 15/2 - 6 = 54/4 - 15/2 - 6 = 6-6 = 0 At x = 2/3: 6(2/3)2 + 5(2/3) - 6 = 6Γ(4/9) + 10/3 - 6 = 24/9 + 10/3 - 6 = 6-6 = 0 Graphing You could also try graphing the quadratic equation. Seeing where it equals zero can give you clues. Example: (continued) Starting with 6x2 + 5x - 6 and just this plot: The roots are around x=β1.5 and x=+0.67, so we can guess the roots are: β3/2 and 2/3 Which can help us work out the factors 2x + 3 and 3x β 2 Always check though! 0.67 might not be 2/3 for example. The General Solution There is also a general solution (useful when the above method fails), which uses the quadratic formula: Quadratic Formula Use that formula to get the two answers x+ and xβ (one is for the + case, and the other is for the β case in the Β±), and we get this factoring: a(x β x+)(x β xβ) Let us use the previous example to see how that works: Example: what are the roots of 6x2 + 5x β 6 ? Substitute a=6, b=5 and c=β6 into the formula: x = (βb Β± β[b2 β 4ac]) / 2a x = (β5 Β± β[52 β 4Γ6Γ(β6)]) / 2Γ6 = (β5 Β± β[25 + 144]) / 12 = (β5 Β± β169) / 12 = (β5 Β± 13) / 12 So the two roots are: x+ = (-5 + 13) / 12 = 8/12 = 2/3, xβ = (-5 β 13) / 12 = β18/12 = β3/2 (Notice that we get the same result we did with the factoring we used before) Now put those values into a(x β x+)(x β xβ): 6(x β 2/3)(x + 3/2) We can rearrange that a little to simplify it: 3(x β 2/3) Γ 2(x + 3/2) = (3x β 2)(2x + 3) And we get the same factors as we did before!
Posted on: Tue, 26 Aug 2014 12:07:29 +0000