Question 800723
The formula given is not for applying blindly to a polynomial.
It is used to apply to a quadratic equation of the form {{{at^2+bt+c =0}}},
meaning an equation that says that a quadratic polynomial {{{highlight(equals)}}} {{{highlight(zero)}}}.
 
To get from your problem to such an equation, whose solutions would be given by
{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
you have to start by stating (in algebra language) that the population equals 1620,
and then do some manipulation to get to the form {{{at^2+bt+c=0}}} with that zero on one side of the equal sign.
 
{{{42t^2+36t+1008=1620}}} --> {{{42t^2+36t+1008-1620=0}}} --> {{{42t^2+36t-612=0}}}
With the {{{-612}}} term without {{{x}}} being opposite in sign to the {{{42}}} coefficient of {{{x^2}}} you are assured to get a positive number under the square root.
Also, at any point in the line above, you could realize that everything is divisible by 6, and dividing by 6 makes the numbers smaller.
For example,
{{{42t^2+36t-612=0}}} --> {{{(42t^2+36t-612)/6=0/6}}} --> {{{(42t^2+36t-612)/6=0/6}}} --> {{{7t^2+6t-102=0}}}
That simplification is not absolutely needed, but helps if you are prone to make mistakes when writing long numbers.
 
Then, you apply the formula, but the minus sign in front of the square root will give you a negative result, which would not make sense for the problem, so the useful result will be calculated as
{{{t=(-36+sqrt(36^2-4*42*(-612)))/(2*42)= (-36+sqrt(1296+102816))/84= (-36+sqrt(104112))/84}}}= approximately {{{(-36+322.66)/84=286.66/84}}}=approximately {{{3.4}}}
or as
{{{t=(-6+sqrt(6^2-4*7*(-102)))/(2*7)= (-6+sqrt(36+2856))/14=(-6+sqrt(2892))/14}}}= approximately {{{(-6+53.78)/84=47.78/14}}}=approximately {{{3.4}}}