SOLUTION: If (a - b)² = 5 and ab = 1, find the positive value of a + b. --- I've successfully solved similar problems with addition of squares, but not a difference of squares.

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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): You can put this solution on YOUR website!


So,


Substituting,



So the positive value is 3.
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
(a - b)^2; = 5 and ab = 1, find the positive value of a + b.
----
b = 1/a
(a - 1/a)^2 = 5
a - 1/a = sqrt(5)
a^2 - a*sqrt(5) - 1 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=9 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 2.61803398874989, -0.381966011250105. Here's your graph:

==================
a = (3 + sqrt(5))/2
b = 2/(3+sqrt(5)) = (3-sqrt(5))/2
a + b = 3
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
(a-b)^2 = 5
:
take square root of both sides
:
1) a - b = square root(5)
:
2) ab = 1
:
solve equation 2 for a
:
a = 1/b
:
substitute for a in equation 1
:
(1/b) - b = square root(5)
:
b^2 +square root(5)b -1 = 0
:
use quadratic formula to find b
:
b = (-square root(5) + square root(5 + 4)) / 2 = (3 - square root(5)) / 2 = 0.382
:
a = 1 / ((3 - square root(5)) / 2) = 2 / (3 - square root(5)) = 2.618
:
*************************
a + b = 0.382 + 2.618 = 3
*************************
:
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~ 

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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