SOLUTION: Two numbers differ by 6. if the numbers are squared and then added, the result is 146. What are the numbers?

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Question 177254: Two numbers differ by 6. if the numbers are squared and then added, the result is 146. What are the numbers?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"Two numbers differ by 6" translates to x-y=6. If we solve for "y", we get y=x-6

"if the numbers are squared and then added, the result is 146" means that x%5E2%2By%5E2=146


x%5E2%2By%5E2=146 Start with the second equation.


x%5E2%2B%28x-6%29%5E2=146 Plug in y=x-6


x%5E2%2Bx%5E2-12x%2B36=146 FOIL


x%5E2%2Bx%5E2-12x%2B36-146=0 Subtract 146 from both sides.


2x%5E2-12x-110=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-12, and c=-110


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-%28-12%29+%2B-+sqrt%28+%28-12%29%5E2-4%282%29%28-110%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-12, and c=-110


x+=+%2812+%2B-+sqrt%28+%28-12%29%5E2-4%282%29%28-110%29+%29%29%2F%282%282%29%29 Negate -12 to get 12.


x+=+%2812+%2B-+sqrt%28+144-4%282%29%28-110%29+%29%29%2F%282%282%29%29 Square -12 to get 144.


x+=+%2812+%2B-+sqrt%28+144--880+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-110%29 to get -880


x+=+%2812+%2B-+sqrt%28+144%2B880+%29%29%2F%282%282%29%29 Rewrite sqrt%28144--880%29 as sqrt%28144%2B880%29


x+=+%2812+%2B-+sqrt%28+1024+%29%29%2F%282%282%29%29 Add 144 to 880 to get 1024


x+=+%2812+%2B-+sqrt%28+1024+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2812+%2B-+32%29%2F%284%29 Take the square root of 1024 to get 32.


x+=+%2812+%2B+32%29%2F%284%29 or x+=+%2812+-+32%29%2F%284%29 Break up the expression.


x+=+%2844%29%2F%284%29 or x+=++%28-20%29%2F%284%29 Combine like terms.


x+=+11 or x+=+-5 Simplify.


So the first numbers are either x+=+11 or x+=+-5


y=x-6 Go back to the isolated equation


y=11-6 Plug in the first solution x=11


y=5 Subtract


So when x+=+11, then y=5

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

y=x-6 Go back to the isolated equation


y=-5-6 Plug in the first solution x=11


y=-11 Subtract


So when x+=+-5, then y=-11


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Answer:


So this means that the two numbers are

11 and 5

OR

-5 and -11