SOLUTION: Find the value of 8ab (a2+b2) when a+b=root 5 and a-b = root 3

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Question 1018213: Find the value of 8ab (a2+b2) when a+b=root 5 and a-b = root 3
Answer by ikleyn(52879) About Me  (Show Source):
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Find the value of 8ab (a2+b2) when a+b=root 5 and a-b = root 3
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a + b = sqrt%285%29,   (1)
a - b = sqrt%283%29.   (2)

By adding equations (1) and (2), you get a = %28sqrt%285%29%2Bsqrt%283%29%29%2F2.

By subtracting (2) from (1), you get b = %28sqrt%285%29-sqrt%283%29%29%2F2.

It implies that a%5E2+%2B+b%5E2 = 1%2F4.%28%28sqrt%285%29%2Bsqrt%283%29%29%5E2+%2B+%28sqrt%285%29-sqrt%283%29%29%5E2%29 = 1%2F4.%285%2B3%2B5-3%29 = 10%2F4 = 2.5.

Next, a*b = 1%2F4.%28sqrt%285%29%2Bsqrt%283%29%29%2A%28sqrt%285%29-sqrt%283%29%29 = 1%2F4.%285-3%29 = 1%2F2.

Therefore, 8ab%2A%28a%5E2%2Bb%5E2%29 = 8%2A%281%2F2%29%2A2.5 = 10.

Answer.  8ab%2A%28a%5E2%2Bb%5E2%29 = 10.