SOLUTION: The difference of two positive numbers is 5 and the sum of their squares is 157. Find the numbers. Please help me with this. I've tried writing an equation out to what it says

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Question 221402: The difference of two positive numbers is 5 and the sum of their squares is 157. Find the numbers.
Please help me with this. I've tried writing an equation out to what it says but I'm completely lost.
Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"The difference of two positive numbers is 5" means that x-y=5 (note: this means that x%3Ey)

Also, because "the sum of their squares is 157" we know that x%5E2%2By%5E2=157


x-y=5 Start with the first equation.


-y=5-x Subtract x from both sides.


y=x-5 Multiply both sides by -1 and rearrange the terms.


x%5E2%2By%5E2=157 Move onto the second equation.


x%5E2%2B%28x-5%29%5E2=157 Plug in y=x-5


x%5E2%2Bx%5E2-10x%2B25=157 FOIL


x%5E2%2Bx%5E2-10x%2B25-157=0 Subtract 157 from both sides.


2x%5E2-10x-132=0 Combine like terms.


Notice that the quadratic 2x%5E2-10x-132 is in the form of Ax%5E2%2BBx%2BC where A=2, B=-10, and C=-132


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-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%282%29%28-132%29+%29%29%2F%282%282%29%29 Plug in A=2, B=-10, and C=-132


x+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%282%29%28-132%29+%29%29%2F%282%282%29%29 Negate -10 to get 10.


x+=+%2810+%2B-+sqrt%28+100-4%282%29%28-132%29+%29%29%2F%282%282%29%29 Square -10 to get 100.


x+=+%2810+%2B-+sqrt%28+100--1056+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-132%29 to get -1056


x+=+%2810+%2B-+sqrt%28+100%2B1056+%29%29%2F%282%282%29%29 Rewrite sqrt%28100--1056%29 as sqrt%28100%2B1056%29


x+=+%2810+%2B-+sqrt%28+1156+%29%29%2F%282%282%29%29 Add 100 to 1056 to get 1156


x+=+%2810+%2B-+sqrt%28+1156+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2810+%2B-+34%29%2F%284%29 Take the square root of 1156 to get 34.


x+=+%2810+%2B+34%29%2F%284%29 or x+=+%2810+-+34%29%2F%284%29 Break up the expression.


x+=+%2844%29%2F%284%29 or x+=++%28-24%29%2F%284%29 Combine like terms.


x+=+11 or x+=+-6 Simplify.


So the possible solutions are x+=+11 or x+=+-6


However, it's stated that the numbers are positive. So the only answer is x=11


Now just plug that value into x-y=5 and solve for 'y' to get 11-y=5 ---> y=11-5=6


So the two numbers are x=11 and y=6


Notice how 11-6=5. So their difference is 5

Also, take note how 11%5E2%2B6%5E2=121%2B36=157.

So this confirms our answer.