Find all possible solutions:
P=(x-1)*(x-2)*(x-4)*(x-8) =ax^2
Solve to get (x,y) for P(a)
Evaluate:
i. (x,y) for a=7 (original version)
ii. for a=4 (more friendly results!)
Credit: Valery Volkov
(x-1)*(x-2)*(x-4)*(x-8) = ax^2
Group the left side and multiply (x-2)*(x-4) together and (x-1)*(x-8) together:
(x^2-9x+8) * (x^2-6x+8) = ax^2
Now split the linear terms to form factors as a sum and difference of expressions:
(x^2-(15/2)x+8 - (3/2)x) * (x^2-(15/2)x+8 + (3/2)x) = ax^2
Now multiply as a difference of squares:
(x^2-(15/2)x+8)^2 - (9/4)x^2 = ax^2
Now just move the x^2 terms to the same side and take square roots:
x^2-(15/2)x+8 = +/- sqrt[a+9/4]*x
At this point we have two very similar quadratic equations. Running through the quadratic formula then we get four similarly structured values for x:
x = (15-sqrt[4a+9]-sqrt[4a+106-30sqrt[4a+9]])/4
x = (15+sqrt[4a+9]-sqrt[4a+106+30sqrt[4a+9]])/4
x = (15-sqrt[4a+9]+sqrt[4a+106-30sqrt[4a+9]])/4
x = (15+sqrt[4a+9]+sqrt[4a+106+30sqrt[4a+9]])/4
If a=4 then the roots are
x = 5/2 - i*sqrt[7]/2
x = 5 - sqrt[17]
x = 5/2 + i*sqrt[7]/2
x = 5 + sqrt[17]
If a=7 then the roots are
x = 15/4 - sqrt[37]/4 - i*sqrt[-134+30sqrt[37]]/4
x = 15/4 + sqrt[37]/4 - sqrt[134+30sqrt[37]]/4
x = 15/4 - sqrt[37]/4 + i*sqrt[-134+30sqrt[37]]/4
x = 15/4 + sqrt[37]/4 + sqrt[134+30sqrt[37]]/4
Bonus if a=10 then the roots are
x = 2 - 2i
x = 11/2 - sqrt[89]/2
x = 2 + 2i
x = 11/2 + sqrt[89]/2
Analytic Solution
