0=10x^2-400x-409

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Solution for 0=10x^2-400x-409 equation:


Simplifying
0 = 10x2 + -400x + -409

Reorder the terms:
0 = -409 + -400x + 10x2

Solving
0 = -409 + -400x + 10x2

Solving for variable 'x'.

Combine like terms: 0 + 409 = 409
409 + 400x + -10x2 = -409 + -400x + 10x2 + 409 + 400x + -10x2

Reorder the terms:
409 + 400x + -10x2 = -409 + 409 + -400x + 400x + 10x2 + -10x2

Combine like terms: -409 + 409 = 0
409 + 400x + -10x2 = 0 + -400x + 400x + 10x2 + -10x2
409 + 400x + -10x2 = -400x + 400x + 10x2 + -10x2

Combine like terms: -400x + 400x = 0
409 + 400x + -10x2 = 0 + 10x2 + -10x2
409 + 400x + -10x2 = 10x2 + -10x2

Combine like terms: 10x2 + -10x2 = 0
409 + 400x + -10x2 = 0

Begin completing the square.  Divide all terms by
-10 the coefficient of the squared term: 

Divide each side by '-10'.
-40.9 + -40x + x2 = 0

Move the constant term to the right:

Add '40.9' to each side of the equation.
-40.9 + -40x + 40.9 + x2 = 0 + 40.9

Reorder the terms:
-40.9 + 40.9 + -40x + x2 = 0 + 40.9

Combine like terms: -40.9 + 40.9 = 0.0
0.0 + -40x + x2 = 0 + 40.9
-40x + x2 = 0 + 40.9

Combine like terms: 0 + 40.9 = 40.9
-40x + x2 = 40.9

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 = 40.9 + 400

Reorder the terms:
400 + -40x + x2 = 40.9 + 400

Combine like terms: 40.9 + 400 = 440.9
400 + -40x + x2 = 440.9

Factor a perfect square on the left side:
(x + -20)(x + -20) = 440.9

Calculate the square root of the right side: 20.997618913

Break this problem into two subproblems by setting 
(x + -20) equal to 20.997618913 and -20.997618913.

Subproblem 1

x + -20 = 20.997618913 Simplifying x + -20 = 20.997618913 Reorder the terms: -20 + x = 20.997618913 Solving -20 + x = 20.997618913 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 = 20.997618913 + 20 Combine like terms: -20 + 20 = 0 0 + x = 20.997618913 + 20 x = 20.997618913 + 20 Combine like terms: 20.997618913 + 20 = 40.997618913 x = 40.997618913 Simplifying x = 40.997618913

Subproblem 2

x + -20 = -20.997618913 Simplifying x + -20 = -20.997618913 Reorder the terms: -20 + x = -20.997618913 Solving -20 + x = -20.997618913 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 = -20.997618913 + 20 Combine like terms: -20 + 20 = 0 0 + x = -20.997618913 + 20 x = -20.997618913 + 20 Combine like terms: -20.997618913 + 20 = -0.997618913 x = -0.997618913 Simplifying x = -0.997618913

Solution

The solution to the problem is based on the solutions from the subproblems. x = {40.997618913, -0.997618913}

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