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.

Algebra ->  Radicals -> 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.      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
%28a-b%29%5E2=a%5E2-2ab%2Bb%5E2
%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2
So,
%28a%2Bb%29%5E2-%28a-b%29%5E2=a%5E2%2B2ab%2Bb%5E2-%28a%5E2-2ab%2Bb%5E2%29
%28a%2Bb%29%5E2-%28a-b%29%5E2=4ab
Substituting,
%28a%2Bb%29%5E2-5=4%281%29
%28a%2Bb%29%5E2=9
a%2Bb=0+%2B-+3
So the positive value is 3.

Answer by Alan3354(69443) About Me  (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 ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2.23606797749979x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2.23606797749979%29%5E2-4%2A1%2A-1=9.

Discriminant d=9 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2.23606797749979%2B-sqrt%28+9+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2.23606797749979%29%2Bsqrt%28+9+%29%29%2F2%5C1+=+2.61803398874989
x%5B2%5D+=+%28-%28-2.23606797749979%29-sqrt%28+9+%29%29%2F2%5C1+=+-0.381966011250105

Quadratic expression 1x%5E2%2B-2.23606797749979x%2B-1 can be factored:
1x%5E2%2B-2.23606797749979x%2B-1+=+%28x-2.61803398874989%29%2A%28x--0.381966011250105%29
Again, the answer is: 2.61803398874989, -0.381966011250105. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2.23606797749979%2Ax%2B-1+%29

==================
a = (3 + sqrt(5))/2
b = 2/(3+sqrt(5)) = (3-sqrt(5))/2
a + b = 3

Answer by rothauserc(4718) About Me  (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(52781) About Me  (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.