How do you factor 15b^2 - 7b - 215b2−7b−2?

1 Answer
Sharkbasket
Mar 27, 2018

(3b-2)(5b+1)(3b−2)(5b+1)

Explanation:

We have a quadratic expression in the form

ax^2+bx+cax2+bx+c

a=15a=15
b=-7b=−7
c=-2c=−2

To factor this, first find two numbers that multiply to give acac and add to give bb.

For ac=-30ac=−30 and b=-7b=−7

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

-10−10 and 33

How did I see that? Look at the factors of 3030

1,2,3,5,6,10,15,301,2,3,5,6,10,15,30

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

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

Now it's much easier to see that -10−10 and 33 are the numbers we want.

So why did we do this? Our original expression was

15b^2-7b-215b2−7b−2

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

rArr15b^2⇒15b2 color(blue)(-10b)−10b color(red)(+3b)-2+3b−2

Now factor 5b5b from the first two terms.

5b(3b-2)+(3b-2)5b(3b−2)+(3b−2)

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

(3b-2)(5b+1)(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⇒15b2 color(red)(+3b)+3b color(blue)(-10b)-2−10b−2

rArr3b(5b+1)-2(5b+1)⇒3b(5b+1)−2(5b+1)

rArr(5b+1)(3b-2)⇒(5b+1)(3b−2)

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