0.00052-x=x^2+0.0040x

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Solution for 0.00052-x=x^2+0.0040x equation:


Simplifying
0.00052 + -1x = x2 + 0.0040x

Reorder the terms:
0.00052 + -1x = 0.0040x + x2

Solving
0.00052 + -1x = 0.0040x + x2

Solving for variable 'x'.

Combine like terms: -1x + -0.0040x = -1.004x
0.00052 + -1.004x + -1x2 = 0.0040x + x2 + -0.0040x + -1x2

Reorder the terms:
0.00052 + -1.004x + -1x2 = 0.0040x + -0.0040x + x2 + -1x2

Combine like terms: 0.0040x + -0.0040x = 0.0000
0.00052 + -1.004x + -1x2 = 0.0000 + x2 + -1x2
0.00052 + -1.004x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
0.00052 + -1.004x + -1x2 = 0

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

Divide each side by '-1'.
-0.00052 + 1.004x + x2 = 0

Move the constant term to the right:

Add '0.00052' to each side of the equation.
-0.00052 + 1.004x + 0.00052 + x2 = 0 + 0.00052

Reorder the terms:
-0.00052 + 0.00052 + 1.004x + x2 = 0 + 0.00052

Combine like terms: -0.00052 + 0.00052 = 0.00000
0.00000 + 1.004x + x2 = 0 + 0.00052
1.004x + x2 = 0 + 0.00052

Combine like terms: 0 + 0.00052 = 0.00052
1.004x + x2 = 0.00052

The x term is 1.004x.  Take half its coefficient (0.502).
Square it (0.252004) and add it to both sides.

Add '0.252004' to each side of the equation.
1.004x + 0.252004 + x2 = 0.00052 + 0.252004

Reorder the terms:
0.252004 + 1.004x + x2 = 0.00052 + 0.252004

Combine like terms: 0.00052 + 0.252004 = 0.252524
0.252004 + 1.004x + x2 = 0.252524

Factor a perfect square on the left side:
(x + 0.502)(x + 0.502) = 0.252524

Calculate the square root of the right side: 0.502517661

Break this problem into two subproblems by setting 
(x + 0.502) equal to 0.502517661 and -0.502517661.

Subproblem 1

x + 0.502 = 0.502517661 Simplifying x + 0.502 = 0.502517661 Reorder the terms: 0.502 + x = 0.502517661 Solving 0.502 + x = 0.502517661 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.502' to each side of the equation. 0.502 + -0.502 + x = 0.502517661 + -0.502 Combine like terms: 0.502 + -0.502 = 0.000 0.000 + x = 0.502517661 + -0.502 x = 0.502517661 + -0.502 Combine like terms: 0.502517661 + -0.502 = 0.000517661 x = 0.000517661 Simplifying x = 0.000517661

Subproblem 2

x + 0.502 = -0.502517661 Simplifying x + 0.502 = -0.502517661 Reorder the terms: 0.502 + x = -0.502517661 Solving 0.502 + x = -0.502517661 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.502' to each side of the equation. 0.502 + -0.502 + x = -0.502517661 + -0.502 Combine like terms: 0.502 + -0.502 = 0.000 0.000 + x = -0.502517661 + -0.502 x = -0.502517661 + -0.502 Combine like terms: -0.502517661 + -0.502 = -1.004517661 x = -1.004517661 Simplifying x = -1.004517661

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.000517661, -1.004517661}

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