Question 1065509: If (a - b)² = 5 and ab = 1, find the positive value of a + b.
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I've successfully solved similar problems with addition of squares, but not a difference of squares.
Found 4 solutions by Fombitz, Alan3354, rothauserc, ikleyn: Answer by Fombitz(32388) (Show Source): Answer by Alan3354(69443) (Show Source): Answer by rothauserc(4718) (Show Source):
You can put this solution on YOUR website! (a-b)^2 = 5
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take square root of both sides
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1) a - b = square root(5)
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2) ab = 1
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solve equation 2 for a
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a = 1/b
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substitute for a in equation 1
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(1/b) - b = square root(5)
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b^2 +square root(5)b -1 = 0
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use quadratic formula to find b
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b = (-square root(5) + square root(5 + 4)) / 2 = (3 - square root(5)) / 2 = 0.382
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a = 1 / ((3 - square root(5)) / 2) = 2 / (3 - square root(5)) = 2.618
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a + b = 0.382 + 2.618 = 3
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Answer by ikleyn(52777) (Show Source):
You can put this solution on YOUR website! .
If (a - b)^2 = 5 and ab = 1, find the positive value of a + b.
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If (a-b)^2 = 2, then a^2 - 2ab + b^2 = 5.
Hensce, a^2 + 2ab + b^2 = (a^2 - 2ab + b^2) + 4ab = 5 + 4 = 9.
In other words, (a+b)^2 = 9.
Then a + b = 3.
It is the same as the Fombitz' solution, but more straightforward.
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