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I notice that almost all of new calculators are able to display the roots of quadratic equations in exact form. For example: 

x^2-16x+14=0
x1=8+5sqrt2
x2=8-5sqrt2

What algorithm could I use to achieve that? I've been searching around but I found no results related to this problem

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Assuming your quadratic equation is in the form 

y = ax^2+bx+c

you get the two roots by

x_1,x_2 = ( -b +- sqrt(b^2-4ac)) / 2a

when for one you use the + between b and the square root, and for the other the -. If you want to take something out of the square root, just compute the factors of the argument and take out the ones with exponent greater than 2.

By the way, the two root you posted are wrong.

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The “algorithm” is exactly the same as on paper. Depending on the programming language, it may start with int delta = b*b - 4*a*c;.

You may want to define a datatype of terms and simplifications on them, though, in case the coefficients of the equation are not simply integer but themselves solutions of previous equations. If this is the sort of thing you are after, look up “symbolic computation”. Some languages are better for this purpose than others. I expect that elementary versions of what you are asking is actually used as an example in some ML tutorials, for instance (see chapter 9).

Pascal Cuoq
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