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Fill the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:

$$3ab - 12a + 7b + \underline{~~~~}$$

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Apr 21, 2020

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Hi guest!

$$3ab - 12a + 7b + \underline{~~~~}$$

So let's start off by noticing the signs. Since all the signs are positive except for -12a, that means that $$a$$ has to be multiplied with a negative number.

This is what we have so far: $$(\text{_}a+\text{_})(\text{_}b-\text{_})$$ (I'm using the blank spaces to represent numbers we don't know yet)

Now, we see that there is a $$7b$$ term. What do we have to multiply $$b$$ by to get $$7b$$? It's $$7$$. So we can put $$7$$ in the blank on the right of $$a$$.

Now we have: $$(\text{_}a+7)(\text{_}b-\text{_})$$

We also know that there is a $$3ab$$ term. Since there is  $$-12a$$ term, we have to multiply $$a$$ by 3 since -12a is divisible by 3.

Now we have: $$(3a+7)(\text{_}b-\text{_})$$

The only term we have to deal with is the $$-12a$$ term. What do we have to multiply $$3a$$ by to get $$-12a$$?  It's $$4$$! So we can put the 4 on the blank.

Now we have: $$(3a+7)(b-4)$$

All we have to do now is multiply it all out!

3a*b=3ab

3a*4=-12a

7*b=7b

7*-4=-28

Adding all the terms together, we see that the quadratic is: $$3ab-12a+7b-28$$

So the blank is $$\boxed{-28}$$

I hope this helped you, guest!
If you have any questions, don't hesitate to ask!

:)

Apr 21, 2020