# SOLUTION: The sum of 2 numbers is 25. The sum of their squares is 313. Find the two numbers...thank you!

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 Question 39164: The sum of 2 numbers is 25. The sum of their squares is 313. Find the two numbers...thank you!Found 3 solutions by Nate, AnlytcPhil, checkley71:Answer by Nate(3500)   (Show Source): You can put this solution on YOUR website! Simplify: Plug in: or Plug in: and The numbers are twelve and thirteen. Answer by AnlytcPhil(1277)   (Show Source): You can put this solution on YOUR website!```The sum of 2 numbers is 25. The sum of their squares is 313. Find the two numbers...thank you! >>...The sum of 2 numbers is 25...<< x + y = 25 >>...The sum of their squares is 313...<< x² + y² = 313 So you have the system of equations: x + y = 25 x² + y² = 313 To solve by substitution: Solve the first equation for y y = 25 - x Substityte (25 - x) for y in x² + y² = 313 x² + (25 - x)² = 313 x² + (625 - 50x + x²) = 313 x² + 625 - 50x + x² = 313 2x² - 50x + 625 = 313 2x² - 50x + 625 = 313 2x² - 50x + 312 = 0 2(x² - 25x + 156) = 0 Factor: two integers which multiply to give 156 and add to give 25 are 13 and 12. 2(x - 13)(x - 12) = 0 Set each factor = 0 and solve: x - 13 = 0 gives x = 13 x - 12 = 0 gives x = 12 Then substitute each of these in y = 25 - x Substituting x = 13 y = 25 - 13 y = 12 So one solution is (x, y) = (13, 12) Substituting x = 12 y = 25 - 12 y = 13 So one solution is (x, y) = (12, 13) Either way the two numbers are 12 and 13. Edwin``` Answer by checkley71(8403)   (Show Source): You can put this solution on YOUR website!X+Y=25 OR & X=25-Y THEN SUBSTITUTE 25-Y FOR X IN EQ #2 WE GET X~2+Y~2=313 OR (25-Y)~2+Y~2=313 OR 625-50Y+Y~2+Y~2=313 OR 2Y~2-50+625=313 OR 2Y~2-50Y+312=0 OR (2Y-26)(Y-12)=0 OR 2Y=26 OR Y=13 & Y=12 THEN X=12 OR X=13 PROOF 12+13=25 & 12~2+13~2=144+169=313 OR 313=313