x^2=400-x

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Solution for x^2=400-x equation:


Simplifying
x2 = 400 + -1x

Solving
x2 = 400 + -1x

Solving for variable 'x'.

Reorder the terms:
-400 + x + x2 = 400 + -1x + -400 + x

Reorder the terms:
-400 + x + x2 = 400 + -400 + -1x + x

Combine like terms: 400 + -400 = 0
-400 + x + x2 = 0 + -1x + x
-400 + x + x2 = -1x + x

Combine like terms: -1x + x = 0
-400 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '400' to each side of the equation.
-400 + x + 400 + x2 = 0 + 400

Reorder the terms:
-400 + 400 + x + x2 = 0 + 400

Combine like terms: -400 + 400 = 0
0 + x + x2 = 0 + 400
x + x2 = 0 + 400

Combine like terms: 0 + 400 = 400
x + x2 = 400

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 400 + 0.25

Reorder the terms:
0.25 + x + x2 = 400 + 0.25

Combine like terms: 400 + 0.25 = 400.25
0.25 + x + x2 = 400.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 400.25

Calculate the square root of the right side: 20.006249024

Break this problem into two subproblems by setting 
(x + 0.5) equal to 20.006249024 and -20.006249024.

Subproblem 1

x + 0.5 = 20.006249024 Simplifying x + 0.5 = 20.006249024 Reorder the terms: 0.5 + x = 20.006249024 Solving 0.5 + x = 20.006249024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 20.006249024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 20.006249024 + -0.5 x = 20.006249024 + -0.5 Combine like terms: 20.006249024 + -0.5 = 19.506249024 x = 19.506249024 Simplifying x = 19.506249024

Subproblem 2

x + 0.5 = -20.006249024 Simplifying x + 0.5 = -20.006249024 Reorder the terms: 0.5 + x = -20.006249024 Solving 0.5 + x = -20.006249024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -20.006249024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -20.006249024 + -0.5 x = -20.006249024 + -0.5 Combine like terms: -20.006249024 + -0.5 = -20.506249024 x = -20.506249024 Simplifying x = -20.506249024

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.506249024, -20.506249024}

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