SOLUTION: Find all real numbers a and b such that a + b = 14 a^3 + b^3 = 812 + a^2 + b^2

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Question 1209837: Find all real numbers a and b such that
a + b = 14
a^3 + b^3 = 812 + a^2 + b^2
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


a%2Bb=14

%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2=196 --> a%5E2%2Bb%5E2=196-2ab

%28a%2Bb%29%5E3=a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3=2744 --> a%5E3%2Bb%5E3=2744-3ab%28a%2Bb%29=2744-42ab

a%5E3%2Bb%5E3=812%2Ba%5E2%2Bb%5E2
2744-42ab=812%2B196-2ab
1736=40ab
ab=217%2F5

a%2814-a%29=217%2F5
14a-a%5E2=217%2F5
a%5E2-14a%2B217%2F5=0

Use the quadratic formula....

a=7%2B2sqrt%2835%29%2F5 or a=7-2sqrt%2835%29%2F5

ANSWER: The two numbers a and b are, in either order, 7%2B2sqrt%2835%29%2F5 and 7-2sqrt%2835%29%2F5