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

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

Guest Apr 21, 2020

#1**+2 **

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!

:)

lokiisnotdead Apr 21, 2020