Question

Factor the following problems SHOW YOUR WORK 1. x^2+16x+24 2. x^2-10a+25

Answer 1

x^2+16x+24:  Think:  What are the integer factors of 24?  I'd list 1, 2, 3, 4, 6, 8, 12 and 24.  Next:   Look for a pair of factors from that list that: 

(1)ADD up to 16 AND (2) whose product is 24.  Unfortunately there seems to be no such pair from the above list.

So:  Use the quadratic formula to find the roots of x^2+16x+24:

        -16 plus or minus sqrt[ 256-4(1)(24) ]
x = --------------------------------------------------
                                  2
          -16 plus or minus sqrt(160)       -16 plus or minus 4 sqrt(10)
    = --------------------------------------- = ----------------------------------------
                                   2                                               2
           2(-8 plus or minus 2 sqrt(10) )
     = ------------------------------------------- = -8 plus or minus 2sqrt(10)
                                2

If c is a root, (x-c) is a factor.  Thus, if -8 plus 2sqrt(10) is a root,

(x+8-2sqrt(10) is a factor.  Can you find the other factor?


x^2-10a+25 is much easier to factor.  The sqrt of 25 is 5, so try x= -5 as a root.  Dividing x-5 into x^2-10a+25, we get x-5.  Therefore, the factors are

(x-5) and (x-5).