Question 962445
start with:


n = aT^2 - 4T + m


subtract m from both sides of the equation to get:


n - m = aT^2 - 4T


factor out the a on the right side of the equation to get:


n - m = a * (T^2 - 4/a * T)


in the preceding step you had to be a little creative because a * -4/a*T = -4*T


divide both sides of the equation by a to get:


(n-m)/a = T^2 - 4/a * T)


complete the squares on the right side of the equation to get:


(n-m)/a = (T-2/a)^2 - 4/a^2


add 4/a^2 to both sides of the equation to get:


(n-m)/a + 4/a^2 = (T-2/a)^2


take the square root of both sides of the equation to get:


sqrt((n-m)/a + 4/a^2) = T - 2/a


add 2/a to both sides of the equation to get:


sqrt((n-m)/a - 4/a^2) + 2/a = T


that's your solution.


T = sqrt((n-m)/a + 4/a^2) + 2/a


the way to check is to give a random value to a and T and m and solve for n in the original equation.


then use that value of n and the values of a and m previously chosen to solve for T in the final equation.


I did and the solution is confirmed as good.


I used:


a = 9
m = 5
t = 2


i solved for n using the original equation to get n = 33.


i then used:


a = 9
m = 5
n = 33


i solved for t using the final equation to get t = 2.


that confirmed the solution is good.