Solve the system
![\[\left\{\begin{matrix} a+b & = &4 \\ a^4+b^4& = & 82 \end{matrix}\right.\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-7bd89430b4e4322cf6e65bacd4d14c07_l3.png "Rendered by QuickLaTeX.com")
Solution
We set 
and 
where ![k \in [-2, 2]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-ad55263d37f5ce260e18300386834e7f_l3.png "Rendered by QuickLaTeX.com")
. The second equation of course is written as
![\[\left ( 2+k \right )^4 + \left ( 2-k \right )^4 = 82 \Leftrightarrow k^4 +24k^2 -25 =0 \Leftrightarrow k = \pm 1\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-0c47d7646ee4fc9d9e0c5b540863700f_l3.png "Rendered by QuickLaTeX.com")
Now 
‘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
![\[\alpha = \frac{\alpha + \beta}{2} + \frac{\alpha-\beta}{2} \quad , \quad \beta = \frac{\alpha + \beta}{2} - \frac{\alpha-\beta}{2}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-b010a3c51315030b5860e11feaddbcb1_l3.png "Rendered by QuickLaTeX.com")
![\[\alpha \beta = \left ( \frac{\alpha + \beta}{2} \right )^2 - \left ( \frac{\alpha - \beta}{2} \right )^2\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-3e47beb792307f473c51422df551526e_l3.png "Rendered by QuickLaTeX.com")
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 
. Try to solve this using the above technique.
The above exercise , along with the historical note , can be found at mathematica.gr .