Solve for x: 4 x + (x - b) (a x + 5) = -10 + c x + 4 x^2
Expand and collect in terms of x: -5 b + x (9 - a b) + a x^2 = -10 + c x + 4 x^2
Subtract -10 + c x + 4 x^2 from both sides: 10 - 5 b + x (9 - a b) - c x - 4 x^2 + a x^2 = 0
Expand and collect in terms of x: 10 - 5 b + x (9 - a b - c) + x^2 (a - 4) = 0
Divide both sides by a - 4: (10 - 5 b)/(a - 4) + (x (9 - a b - c))/(a - 4) + x^2 = 0
Subtract (10 - 5 b)/(a - 4) from both sides: (x (9 - a b - c))/(a - 4) + x^2 = -(10 - 5 b)/(a - 4)
Add (9 - a b - c)^2/(4 (a - 4)^2) to both sides: (9 - a b - c)^2/(4 (a - 4)^2) + (x (9 - a b - c))/(a - 4) + x^2 = (9 - a b - c)^2/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)
Write the left hand side as a square: ((9 - a b - c)/(2 (a - 4)) + x)^2 = (9 - a b - c)^2/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)
Take the square root of both sides: (9 - a b - c)/(2 (a - 4)) + x = sqrt((9 - a b - c)^2/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)) or (9 - a b - c)/(2 (a - 4)) + x = -sqrt((9 - a b - c)^2/(4 (a - 4)^2) - (10 - 5 b)/(a - 4))
Subtract (9 - a b - c)/(2 (a - 4)) from both sides: x = sqrt(((9 - a b - c)^2)/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)) - (9 - a b - c)/(2 (a - 4)) or (9 - a b - c)/(2 (a - 4)) + x = -sqrt((9 - a b - c)^2/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)) Subtract (9 - a b - c)/(2 (a - 4)) from both sides: Answer: x = sqrt(((9 - a b - c)^2)/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)) - (9 - a b - c)/(2 (a - 4)) or x = -sqrt(((9 - a b - c)^2)/(4 (a - 4)^2) - (10 - 5 b)/(a - 4)) - (9 - a b - c)/(2 (a - 4))
