SOLUTION: Given a+b=1, a^3 + b^3=16 and (a+b)^3 = a^3 +3a^2b +3ab^2 + b^3, find the value of a^2 + b^2.

Algebra.Com
Question 1116399: Given a+b=1, a^3 + b^3=16 and (a+b)^3 = a^3 +3a^2b +3ab^2 + b^3, find the value of a^2 + b^2.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
a+b=1
a^3 + b^3=16
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Sub 1-a for b
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a^3 + (1-a)^3 = 16
a^3 + 1 -3a +3a^2 -a^3 = 16
3a^2 - 3a - 15 = 0
a^2 - a - 5 = 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=21 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 2.79128784747792, -1.79128784747792. Here's your graph:

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a = x1, b = x2 or vice versa
You do a^2 + b^2
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You'll find a^2 + b^2 = 11, so there is more than one way to do it.
Answer by ikleyn(52810)   (Show Source): You can put this solution on YOUR website!
.

The way  (= the ONLY way;  == the CANONICAL way) of solving this problem is  THIS:



1.  1 =  =  =  +  = (replace a^3 + b^3 by 16 and replace a+b by 1, since it is given) = 


        16 + 3ab,

    which implies  3ab = 1 - 16 = -15  and hence  ab = -5.    (*)



2.  Now the next and the last step is  


    1 =  =  =  +  = (replace ab by -5, since we just found it in (*)) =  -  =  - 10,


    which implies   = 1 + 10 = 11.


Answer. Under given conditions,   = 11.


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