100x+x^2=5000

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Solution for 100x+x^2=5000 equation:


Simplifying
100x + x2 = 5000

Solving
100x + x2 = 5000

Solving for variable 'x'.

Reorder the terms:
-5000 + 100x + x2 = 5000 + -5000

Combine like terms: 5000 + -5000 = 0
-5000 + 100x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5000' to each side of the equation.
-5000 + 100x + 5000 + x2 = 0 + 5000

Reorder the terms:
-5000 + 5000 + 100x + x2 = 0 + 5000

Combine like terms: -5000 + 5000 = 0
0 + 100x + x2 = 0 + 5000
100x + x2 = 0 + 5000

Combine like terms: 0 + 5000 = 5000
100x + x2 = 5000

The x term is 100x.  Take half its coefficient (50).
Square it (2500) and add it to both sides.

Add '2500' to each side of the equation.
100x + 2500 + x2 = 5000 + 2500

Reorder the terms:
2500 + 100x + x2 = 5000 + 2500

Combine like terms: 5000 + 2500 = 7500
2500 + 100x + x2 = 7500

Factor a perfect square on the left side:
(x + 50)(x + 50) = 7500

Calculate the square root of the right side: 86.602540378

Break this problem into two subproblems by setting 
(x + 50) equal to 86.602540378 and -86.602540378.

Subproblem 1

x + 50 = 86.602540378 Simplifying x + 50 = 86.602540378 Reorder the terms: 50 + x = 86.602540378 Solving 50 + x = 86.602540378 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-50' to each side of the equation. 50 + -50 + x = 86.602540378 + -50 Combine like terms: 50 + -50 = 0 0 + x = 86.602540378 + -50 x = 86.602540378 + -50 Combine like terms: 86.602540378 + -50 = 36.602540378 x = 36.602540378 Simplifying x = 36.602540378

Subproblem 2

x + 50 = -86.602540378 Simplifying x + 50 = -86.602540378 Reorder the terms: 50 + x = -86.602540378 Solving 50 + x = -86.602540378 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-50' to each side of the equation. 50 + -50 + x = -86.602540378 + -50 Combine like terms: 50 + -50 = 0 0 + x = -86.602540378 + -50 x = -86.602540378 + -50 Combine like terms: -86.602540378 + -50 = -136.602540378 x = -136.602540378 Simplifying x = -136.602540378

Solution

The solution to the problem is based on the solutions from the subproblems. x = {36.602540378, -136.602540378}

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