SOLUTION: The sum of two numbers is 8. The sum of the squares of the two numbers is 34. Find the numbers. I know they are 3 and 5, but how do I show I arrived at that answer

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: The sum of two numbers is 8. The sum of the squares of the two numbers is 34. Find the numbers. I know they are 3 and 5, but how do I show I arrived at that answer      Log On


   

Question 144981: The sum of two numbers is 8. The sum of the squares of the two numbers is 34. Find the numbers.
I know they are 3 and 5, but how do I show I arrived at that answer
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"The sum of two numbers is 8" translates to x%2By=8


"The sum of the squares of the two numbers is 34" translates to x%5E2%2By%5E2=34


x%2By=8 Start with the first equation


y=8-x Subtract x from both sides to isolate y


x%5E2%2By%5E2=34 Move onto the second equation


x%5E2%2B%288-x%29%5E2=34 Plug in y=8-x


x%5E2%2B64-16x%2Bx%5E2=34 Foil


x%5E2%2B64-16x%2Bx%5E2-34=0 Get all terms to the left side.


2x%5E2-16x%2B30=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-16, and c=30


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-%28-16%29+%2B-+sqrt%28+%28-16%29%5E2-4%282%29%2830%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-16, and c=30


x+=+%2816+%2B-+sqrt%28+%28-16%29%5E2-4%282%29%2830%29+%29%29%2F%282%282%29%29 Negate -16 to get 16.


x+=+%2816+%2B-+sqrt%28+256-4%282%29%2830%29+%29%29%2F%282%282%29%29 Square -16 to get 256.


x+=+%2816+%2B-+sqrt%28+256-240+%29%29%2F%282%282%29%29 Multiply 4%282%29%2830%29 to get 240


x+=+%2816+%2B-+sqrt%28+16+%29%29%2F%282%282%29%29 Subtract 240 from 256 to get 16


x+=+%2816+%2B-+sqrt%28+16+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2816+%2B-+4%29%2F%284%29 Take the square root of 16 to get 4.


x+=+%2816+%2B+4%29%2F%284%29 or x+=+%2816+-+4%29%2F%284%29 Break up the expression.


x+=+%2820%29%2F%284%29 or x+=++%2812%29%2F%284%29 Combine like terms.


x+=+5 or x+=+3 Simplify.


So our answers are x+=+5 or x+=+3