How do you factor completely 3x^2 - 5x - 2?

1 Answer
Alan P.
Jun 23, 2016

3x^2-5x-2=color(blue)((1x-2)(3x+1))

Explanation:

Looking for {a,b,c,d} for a factoring (ax+c)(bx+d)=3x^2-5x-2

We see that
color(white)("XXX")ab=3
color(white)("XXX")cd=-2
and
color(white)("XXX")ad+cb=-5

In the hopes of finding integer values for {a,b,c,d}
we note that the factors of 3 are {1,3}
so if a, b are integers {a,b}={1,3}
and
we also note the the factors of -2 are {(-1,2),(1,-2)}

We can test these possible combinations looking for ad+cb=5

{: (underline(a),underline(b),underline(c),underline(d),,underline(ad+cb)), (1,3,-1,2,,-1), (1,3,2,-1,,+5), (1,3,1,-2,,+1), (color(red)(1),color(red)(3),color(red)(-2),color(red)(1),,color(red)(-5)) :}

We have found values that match our requirements:
color(white)("XXX")< a,b,c,d > = < 1,3,-2,1 >
So
color(white)("XXX")(ax+c)(bx+d)=(1x-2)(3x+1)

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