x=40x^2-400x+500

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Solution for x=40x^2-400x+500 equation:


Simplifying
x = 40x2 + -400x + 500

Reorder the terms:
x = 500 + -400x + 40x2

Solving
x = 500 + -400x + 40x2

Solving for variable 'x'.

Reorder the terms:
-500 + x + 400x + -40x2 = 500 + -400x + 40x2 + -500 + 400x + -40x2

Combine like terms: x + 400x = 401x
-500 + 401x + -40x2 = 500 + -400x + 40x2 + -500 + 400x + -40x2

Reorder the terms:
-500 + 401x + -40x2 = 500 + -500 + -400x + 400x + 40x2 + -40x2

Combine like terms: 500 + -500 = 0
-500 + 401x + -40x2 = 0 + -400x + 400x + 40x2 + -40x2
-500 + 401x + -40x2 = -400x + 400x + 40x2 + -40x2

Combine like terms: -400x + 400x = 0
-500 + 401x + -40x2 = 0 + 40x2 + -40x2
-500 + 401x + -40x2 = 40x2 + -40x2

Combine like terms: 40x2 + -40x2 = 0
-500 + 401x + -40x2 = 0

Begin completing the square.  Divide all terms by
-40 the coefficient of the squared term: 

Divide each side by '-40'.
12.5 + -10.025x + x2 = 0

Move the constant term to the right:

Add '-12.5' to each side of the equation.
12.5 + -10.025x + -12.5 + x2 = 0 + -12.5

Reorder the terms:
12.5 + -12.5 + -10.025x + x2 = 0 + -12.5

Combine like terms: 12.5 + -12.5 = 0.0
0.0 + -10.025x + x2 = 0 + -12.5
-10.025x + x2 = 0 + -12.5

Combine like terms: 0 + -12.5 = -12.5
-10.025x + x2 = -12.5

The x term is -10.025x.  Take half its coefficient (-5.0125).
Square it (25.12515625) and add it to both sides.

Add '25.12515625' to each side of the equation.
-10.025x + 25.12515625 + x2 = -12.5 + 25.12515625

Reorder the terms:
25.12515625 + -10.025x + x2 = -12.5 + 25.12515625

Combine like terms: -12.5 + 25.12515625 = 12.62515625
25.12515625 + -10.025x + x2 = 12.62515625

Factor a perfect square on the left side:
(x + -5.0125)(x + -5.0125) = 12.62515625

Calculate the square root of the right side: 3.553189588

Break this problem into two subproblems by setting 
(x + -5.0125) equal to 3.553189588 and -3.553189588.

Subproblem 1

x + -5.0125 = 3.553189588 Simplifying x + -5.0125 = 3.553189588 Reorder the terms: -5.0125 + x = 3.553189588 Solving -5.0125 + x = 3.553189588 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.0125' to each side of the equation. -5.0125 + 5.0125 + x = 3.553189588 + 5.0125 Combine like terms: -5.0125 + 5.0125 = 0.0000 0.0000 + x = 3.553189588 + 5.0125 x = 3.553189588 + 5.0125 Combine like terms: 3.553189588 + 5.0125 = 8.565689588 x = 8.565689588 Simplifying x = 8.565689588

Subproblem 2

x + -5.0125 = -3.553189588 Simplifying x + -5.0125 = -3.553189588 Reorder the terms: -5.0125 + x = -3.553189588 Solving -5.0125 + x = -3.553189588 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.0125' to each side of the equation. -5.0125 + 5.0125 + x = -3.553189588 + 5.0125 Combine like terms: -5.0125 + 5.0125 = 0.0000 0.0000 + x = -3.553189588 + 5.0125 x = -3.553189588 + 5.0125 Combine like terms: -3.553189588 + 5.0125 = 1.459310412 x = 1.459310412 Simplifying x = 1.459310412

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.565689588, 1.459310412}

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