The equation that we must factor is x^2-3x-18!

 

As we can see, when we fill out the equation is algebra tile form we can see that the algebra tiles are not completely completed. There is a large gap that is missing! This is could be from a possible positive – negative equation that caused a section of the tiles to be cancelled out.

 

Now we must find out what that other missing number and the other hidden numbers (that are hiding within these numbers) are. As we can see, the (-)18 is a negative number which would mean that the equation to make this number would have to be a (+) positive and (-) negative.

 

The -18 represents the product of all numbers which would mean that it would be (+)____ x (-)____. And the -3x represents the sum of all numbers which would be (+)____ + (-)_____.

 

in order to find the numbers that when multiplied equal -18 and when added  equal -3, I made a list of all possible numbers that can multiply into -18. Once I had them all down I then added them all together to see which ones would equal -3. I finally found that -6 and 3 both qual -18 when multiplied and -3 when added!

 

So now that we know that our numbers are x^2,-6,3 and 18. We can the complete the table!

 

now all we have to do is fill to the outside of the grid with the applicable algebra tiles! We know that a (+)(+)=(+), that (-)(+)=(-), that(-)(-)=(+), that (x)(x) equals x^2, that (x)(1)=1x and that (1)(1)=1!

 

In the end we end up with (x-6)(x+3) as the factor of x^2-3-18!

 

This shows pattern because in every equation you always have negatives and negatives equal positive, positive and positive equals positive and negative and positive equals negative. Also, with factoring we see that you must always find the sum of all numbers to equal a number and a variable (ex/4x) and the product of all numbers to equal the lonesome number (ex/12). Also with the algebra tiles all the small squares in the bottom right corner along with the fact that the majority of negative tiles are on the lower half of the grid!