40x-x^2=40

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Solution for 40x-x^2=40 equation:


Simplifying
40x + -1x2 = 40

Solving
40x + -1x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + 40x + -1x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + 40x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
40 + -40x + x2 = 0

Move the constant term to the right:

Add '-40' to each side of the equation.
40 + -40x + -40 + x2 = 0 + -40

Reorder the terms:
40 + -40 + -40x + x2 = 0 + -40

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

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

The x term is -40x.  Take half its coefficient (-20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
-40x + 400 + x2 = -40 + 400

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

Combine like terms: -40 + 400 = 360
400 + -40x + x2 = 360

Factor a perfect square on the left side:
(x + -20)(x + -20) = 360

Calculate the square root of the right side: 18.973665961

Break this problem into two subproblems by setting 
(x + -20) equal to 18.973665961 and -18.973665961.

Subproblem 1

x + -20 = 18.973665961 Simplifying x + -20 = 18.973665961 Reorder the terms: -20 + x = 18.973665961 Solving -20 + x = 18.973665961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + x = 18.973665961 + 20 Combine like terms: -20 + 20 = 0 0 + x = 18.973665961 + 20 x = 18.973665961 + 20 Combine like terms: 18.973665961 + 20 = 38.973665961 x = 38.973665961 Simplifying x = 38.973665961

Subproblem 2

x + -20 = -18.973665961 Simplifying x + -20 = -18.973665961 Reorder the terms: -20 + x = -18.973665961 Solving -20 + x = -18.973665961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + x = -18.973665961 + 20 Combine like terms: -20 + 20 = 0 0 + x = -18.973665961 + 20 x = -18.973665961 + 20 Combine like terms: -18.973665961 + 20 = 1.026334039 x = 1.026334039 Simplifying x = 1.026334039

Solution

The solution to the problem is based on the solutions from the subproblems. x = {38.973665961, 1.026334039}

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