<s>[INST] Finding double root. An easier way? Given the polynomial $f = X^4 - 6X^3 + 13X^2 + aX + b$ you have to find the values of $a$ and $b$ such that $f$ has two double roots.
I went about this by writing the polynomial as:
$$f = X^4 - 6X^3 + 13X^2 + aX + b = (X - r)^2(X - s)^2$$
... and from there I arrived at a system of four equations from where I could find the desired values. However, there's quite a bit of work to do it this way; is there a shortcut? 
 [/INST] HINT $\ $ Simpler, first solve $\rm\ f\: =\: (x^2 + c\ x + d)^2\:.\:$ This yields  
$$\rm x^4 - 6\ x^3 +13\ x^2 +\cdots\ =\ x^4 + 2\:c\ x^3 + (2\:d+c^2)\ x^2\:+\cdots$$
hence $\rm\ 2\:c = -6\ $ so $\rm\ c = -3\:.\ $ And $\rm\ 2\:d+c^2 = 2\:d+9 = 13\ $ so $\rm\ d = 2\:.$
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