SOLUTION: Hello, I stumbled on a Algebra problem in my Algebra Problem book today. The problem is: Joe thought of two numbers, their sum was 20 and the sum of their squares was 300. Wha

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Question 365171: Hello, I stumbled on a Algebra problem in my Algebra Problem book today.
The problem is:
Joe thought of two numbers, their sum was 20 and the sum of their squares was 300. What is their product?
I am guessing you have to find all the sums of 20 first?
But the "sum of their squares is 300" part is confusing me.
Is it the two numbers, squared, and the squares then added to receive 300?
WOuld the numbers be negative?
Help would be appreciated, these are the kinds of algebra problems I will require to know for the upcoming quiz.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Joe thought of two numbers, their sum was 20 and the sum of their squares was 300. What is their product?
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Use x and y
x + y = 20
x%5E2+%2B+y%5E2+=+300
Sub for y: y = 20-x
x%5E2+%2B+%2820-x%29%5E2+=+300
2x%5E2+-+40x+%2B+100+=+0
x%5E2+-+20x+%2B+50+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-20x%2B50+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-20%29%5E2-4%2A1%2A50=200.

Discriminant d=200 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--20%2B-sqrt%28+200+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-20%29%2Bsqrt%28+200+%29%29%2F2%5C1+=+17.0710678118655
x%5B2%5D+=+%28-%28-20%29-sqrt%28+200+%29%29%2F2%5C1+=+2.92893218813452

Quadratic expression 1x%5E2%2B-20x%2B50 can be factored:
1x%5E2%2B-20x%2B50+=+%28x-17.0710678118655%29%2A%28x-2.92893218813452%29
Again, the answer is: 17.0710678118655, 2.92893218813452. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-20%2Ax%2B50+%29

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x = 10 ± 5sqrt(2)
--> x = 10 + 5sqrt(2), y = 10 - 5sqrt(2) or vice versa
The product = 100 - 50 = 50