For parabolic equations, finding solutions to the equations can be done by two ways:
(1) Finding multiplicative factors of the constant or the rightmost term (i.e. #-609#) that add up to the numerical value of the middle term (#-8#). This method is recommended if the rightmost term doesn't have much factors. This would be difficult if the rightmost term were rational with several possible factors. Furthermore, this method requires a bit of a trail-and-error.
After multiple attempts of trial-and-error, I have found that
#(x+21)(x-29)=0#
Wherein #609# is the product of factors #-29# and #21# and consequently the sum of these two factors is #-8#.
(2) The quadratic equation.
The quadratic equation is #x=frac(-b+-\sqrt(b^2-4ac))(2a)#.
So what are the variables #a,b,# and #c#? These are just the numerical values of each term in the quadratic equation.
Specifically;
#a# corresponds to the numerical value of the leftmost term;
#b# corresponds to the numerical value of the middle term; and
#c# corresponds to the numerical value of the rightmost term.
From the original equation,
#a=1#
#b=-8#
#c=-609#
Then, plug-in these values into the quadratic equation:
(i) #x=frac(-(-8)+\sqrt((-8)^2-(4)(1)(-609)))(2(1))=29#
(ii) #x=frac(-(-8)-\sqrt((-8)^2-(4)(1)(-609)))(21)=-21#
Therefore,
From the obtained #x# values, the factors should
#(x+21)(x-29)=0#
to maintain the equality of the right side of the equation, #0#.
