There are general solutions for quintic polynomials –
polynomials of order 4. They may be real or not real depending on the
polynomial. We are interested here in a special class of quintic
polynomials which factorises into two quadratics which we can solve.
For example, solve
Substituteso
thatand
the equation becomesThis
factorises to givesoor
4, henceor
4 soor
Example: Solve
Substituteto
getThis
factorises to givehence
orUsing
the substitutionwe
haveorhencewhich
is impossible orThe
only solutions are
Sometimes you have to be sure that you are square rooting a
positive number.
Example
This expression does not factorise but we can use the normal
quadratic formula to solve for
then if the solutions forare
positive, we can square root to obtain
In the equation
Calculation
of these two decimals confirms they are both positive. Hence we can
square root them andor
Example
In the equation
Calculation
of these two decimals confirms they are both negative. Hence we
cannot square root them there are no real roots for this equation.
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