SOLUTION: Hi! What is the process to convert a FOIL expression back into quadratic equation? example: 5p squared + 14pq -3q squared=0 I know that the answer is (5p-q) (p + 3q)

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: Hi! What is the process to convert a FOIL expression back into quadratic equation? example: 5p squared + 14pq -3q squared=0 I know that the answer is (5p-q) (p + 3q)       Log On


   

Question 281637: Hi!
What is the process to convert a FOIL expression back into quadratic equation?
example:
5p squared + 14pq -3q squared=0
I know that the answer is (5p-q) (p + 3q)
What are the operations to arrive at the above expression?
Thank you.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 5p%5E2%2B14pq-3q%5E2 we can see that the first term is 5p%5E2 and the last term is -3q%5E2 where the coefficients are 5 and -3 respectively.

Now multiply the first coefficient 5 and the last coefficient -3 to get -15. Now what two numbers multiply to -15 and add to the middle coefficient 14? Let's list all of the factors of -15:



Factors of -15:
1,3,5,15

-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -15
(1)*(-15)
(3)*(-5)
(-1)*(15)
(-3)*(5)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 14


First NumberSecond NumberSum
1-151+(-15)=-14
3-53+(-5)=-2
-115-1+15=14
-35-3+5=2





From this list we can see that -1 and 15 add up to 14 and multiply to -15


Now looking at the expression 5p%5E2%2B14pq-3q%5E2, replace 14pq with -pq%2B15pq (notice -pq%2B15pq adds up to 14pq. So it is equivalent to 14pq)

5p%5E2%2Bhighlight%28-pq%2B15pq%29-3q%5E2


Now let's factor 5p%5E2-pq%2B15pq-3q%5E2 by grouping:


%285p%5E2-pq%29%2B%2815pq-3q%5E2%29 Group like terms


p%285p-q%29%2B3q%285p-q%29 Factor out the GCF of p out of the first group. Factor out the GCF of 3q out of the second group


%28p%2B3q%29%285p-q%29 Since we have a common term of 5p-q, we can combine like terms

So 5p%5E2-pq%2B15pq-3q%5E2 factors to %28p%2B3q%29%285p-q%29


So this also means that 5p%5E2%2B14pq-3q%5E2 factors to %28p%2B3q%29%285p-q%29 (since 5p%5E2%2B14pq-3q%5E2 is equivalent to 5p%5E2-pq%2B15pq-3q%5E2)



------------------------------------------------------------



Answer:
So 5p%5E2%2B14pq-3q%5E2 factors to %28p%2B3q%29%285p-q%29


In other words, 5p%5E2%2B14pq-3q%5E2=%28p%2B3q%29%285p-q%29