How do you factor #15b^2 - 7b - 2#?

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Answer 1 · 2018-03-27T22:01:50

#(3b-2)(5b+1)#

We have a quadratic expression in the form

#ax^2+bx+c#

#a=15#
#b=-7#
#c=-2#

To factor this, first find two numbers that multiply to give #ac# and add to give #b#.

For #ac=-30# and #b=-7#

Maybe you can see by inspection that these two numbers are:

#-10# and #3#

How did I see that? Look at the factors of #30#

#1,2,3,5,6,10,15,30#

Now try adding or subtracting them to get #-7#

Remember that one of them (and ONLY one of them) must be negative so that we get #-30# when we multiply them.

Now it's much easier to see that #-10# and #3# are the numbers we want.

So why did we do this? Our original expression was

#15b^2-7b-2#

Let's now split the middle term using the numbers we just found.

#rArr15b^2# #color(blue)(-10b)# #color(red)(+3b)-2#

Now factor #5b# from the first two terms.

#5b(3b-2)+(3b-2)#

Notice now that we have a common factor of #(3b-2)#.
Let's factor it out:

#(3b-2)(5b+1)#

And we're done!

NOTE:

It still would have worked if we had chosen to split the terms in the opposite order. Let's check:

#rArr15b^2# #color(red)(+3b)# #color(blue)(-10b)-2#

#rArr3b(5b+1)-2(5b+1)#

#rArr(5b+1)(3b-2)#