Question 152065This question is from textbook Algebra 1 An Incremental Development
: Factor: 6x^2 + x - 2
This question is from textbook Algebra 1 An Incremental Development
Answer by nabla(475)   (Show Source):
You can put this solution on YOUR website! 6x^2 + x - 2=(3x+2)(2x-1)
Explanation:
Consider:
ax^2+bx+c
If we want to factor into form
(dx+e)(fx+g)
, we will need df=a, fe+gd=b, and eg=c.
So, for our problem we have
a=6, b=1, c=-2
So we need df=6, fe+gd=1, eg=-2
Note that d and f must exist in the set A={1,2,3,6}. If we suspect 2 and 3 as d and f, we must must have (1) 2e+3g=1 or (2) 2g+3e=1 with eg still -2.
However, the same rule applies as with df. eg must have factors existing in B={1,2}. We note that -1(2)=-2 and 1(-2)=-2. Therefore these two are possible values of e and g. However, we must decide which is which.
For (1) If e=-1 and g=2, -2+6=4 is not equal to 1. If e=1 and g=-2, 2-6=-4 is not equal to 1.
For (2) If e=-1 and g=2, 4-3=1 is equal to 1. These must be our values.
Thus we have, e=-1, g=2, d=2, f=3, and the factorization is (2x-1)(3x+2).
NOTE: This explanation is quite complicated. You will see with some practice that all of the work I displayed here is totally unnecessary. You will be able to factor out of intuition.
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