  A method is proposed with which the locations of the roots of the monic
symbolic quintic polynomial $x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$
can be determined using the roots of two resolvent quadratic polynomials:
$q_1(x) = x^2 + a_4 x + a_3$ and $q_2(x) = a_2 x^2 + a_1 x + a_0$, whose
coefficients are exactly those of the quintic polynomial. The different cases
depend on the coefficients of $q_1(x)$ and $q_2(x)$ and on some specific
relationships between them. The method is illustrated with the full analysis of
one of the possible cases. Some of the roots of the symbolic quintic equation
for this case have their isolation intervals determined and, as this cannot be
done for all roots with the help of quadratic equations only, finite intervals
containing 1 or 3 roots, or 0 or 2 roots, or, rarely, 0, or 2, or 4 roots of
the quintic are identified. Knowing the stationary points of the quintic
polynomial, lifts the latter indeterminacy and allows one to find the isolation
interval of each of the roots of the quintic. Separately, using the complete
root classification of the quintic, one can also lift this indeterminacy. The
method also allows to see how variation of the individual coefficients of the
quintic affect its roots. No root finding iterations or any numerical
approximations are used and no equations of degree higher than 2 are solved.
