SOLUTION: The sum of two numbers is 32. The sum of the squares of the two numbers is 544. Find the numbers. Thank you for your help, <3 Rae

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Question 251552: The sum of two numbers is 32. The sum of the squares of the two numbers is 544. Find the numbers.
Thank you for your help,
<3 Rae
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"The sum of two numbers is 32" translates to x%2By=32


"The sum of the squares of the two numbers is 544" means that x%5E2%2By%5E2=544


x%2By=32 Start with the first equation.


y=32-x Subtract x from both sides.


x%5E2%2By%5E2=544 Move onto the next equation.


x%5E2%2B%2832-x%29%5E2=544 Plug in y=32-x


x%5E2%2B1024-64x%2Bx%5E2=544 FOIL


x%5E2%2B1024-64x%2Bx%5E2-544=0 Subtract 544 from both sides.


2x%5E2-64x%2B480=0 Combine like terms.


Notice that the quadratic 2x%5E2-64x%2B480 is in the form of Ax%5E2%2BBx%2BC where A=2, B=-64, and C=480


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-64%29+%2B-+sqrt%28+%28-64%29%5E2-4%282%29%28480%29+%29%29%2F%282%282%29%29 Plug in A=2, B=-64, and C=480


x+=+%2864+%2B-+sqrt%28+%28-64%29%5E2-4%282%29%28480%29+%29%29%2F%282%282%29%29 Negate -64 to get 64.


x+=+%2864+%2B-+sqrt%28+4096-4%282%29%28480%29+%29%29%2F%282%282%29%29 Square -64 to get 4096.


x+=+%2864+%2B-+sqrt%28+4096-3840+%29%29%2F%282%282%29%29 Multiply 4%282%29%28480%29 to get 3840


x+=+%2864+%2B-+sqrt%28+256+%29%29%2F%282%282%29%29 Subtract 3840 from 4096 to get 256


x+=+%2864+%2B-+sqrt%28+256+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2864+%2B-+16%29%2F%284%29 Take the square root of 256 to get 16.


x+=+%2864+%2B+16%29%2F%284%29 or x+=+%2864+-+16%29%2F%284%29 Break up the expression.


x+=+%2880%29%2F%284%29 or x+=++%2848%29%2F%284%29 Combine like terms.


x+=+20 or x+=+12 Simplify.


So the solutions are x+=+20 or x+=+12

If x=20, then y=32-20=12 or if x=12 then y=32-12=20.


This means that the two numbers are simply 20 and 12 (in either order).