Non linear system

Solve the system

    \[\left\{\begin{matrix} a+b & = &4 \\ a^4+b^4& = & 82 \end{matrix}\right.\]

Solution

We set a=2+k and b=2-k where k \in [-2, 2]. The second equation of course is written as

    \[\left ( 2+k \right )^4 + \left ( 2-k \right )^4 = 82 \Leftrightarrow k^4 +24k^2 -25 =0 \Leftrightarrow k = \pm 1\]

Now a, b‘s follow.

Historical note: In the Babylonian signs , tracing back in 1700 BC , there are a lot of geometrical problems that are equivalent to the solution of such systems. In order to be solved the following ( modern ) formulae were used

    \[\alpha = \frac{\alpha + \beta}{2} + \frac{\alpha-\beta}{2} \quad , \quad \beta = \frac{\alpha + \beta}{2} - \frac{\alpha-\beta}{2}\]

    \[\alpha \beta = \left ( \frac{\alpha + \beta}{2} \right )^2 - \left ( \frac{\alpha - \beta}{2} \right )^2\]

The above procedure actually led to the discriminant in order for us to solve a second order equation. For example in a book of that age we see the equation x(6+x)=16. Try to solve this using the above technique.

The above exercise , along with the historical note , can be found at mathematica.gr .

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