Question 1095268: how much xsquare -y square ? if x+y=66 xy=9
Found 4 solutions by MathTherapy, greenestamps, ikleyn, KMST:
Answer by MathTherapy(10801)   (Show Source):
Answer by greenestamps(13326)  (Show Source):
Answer by ikleyn(53742)  (Show Source):
Answer by KMST(5344)  (Show Source):
You can put this solution on YOUR website! ONE WAY:
Maybe we know about solving quadratic equations.
Sometimes we can solve a quadratic equation such as  by factoring if we find two integers   such that  and  .
Then those values for and are the solutions of the equation and the equation is really
 .
The solutions to a quadratic equation of the form are always two numbers whose sum and product can be found are  and  respectively.
Unfortunately, sometimes those numbers are irrational, or even imaginary, and then factoring is not an option.
Then, w must use algebra to "complete the square" and then solve, or apply the dreaded quadratic formula
that says the solutions to an equation of the form  are given by
To avoid confusion, I used  for the variable instead of  , and I prefer equations where the leading coefficient is  , so I wrote my equation as .
I will make  and  to get  .
The solutions will be the numbers and (or and  ) that add up to  and whose product is  .
The quadratic formula tells me that the solution are given by
The solutions to the equation are  and 
One is and the other is , but there is no way to guess which was intended to be and which was intended to be .
 and  are
 and
The possible answers are
 and 
ANOTHER APPROACH:
We know that  . If we only knew the value of  we could easily find the value of 
We know that  , but to calculate  we would need the value of 
We know that  and we know the values of  and 
Substituting the known values we get  -->  --> 
Now we can use that value of to calculate the values of and , and from that find the value of
Then 
Multiplying times 
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