x(x+1)=160000

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Solution for x(x+1)=160000 equation:


Simplifying
x(x + 1) = 160000

Reorder the terms:
x(1 + x) = 160000
(1 * x + x * x) = 160000
(1x + x2) = 160000

Solving
1x + x2 = 160000

Solving for variable 'x'.

Reorder the terms:
-160000 + 1x + x2 = 160000 + -160000

Combine like terms: 160000 + -160000 = 0
-160000 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '160000' to each side of the equation.
-160000 + 1x + 160000 + x2 = 0 + 160000

Reorder the terms:
-160000 + 160000 + 1x + x2 = 0 + 160000

Combine like terms: -160000 + 160000 = 0
0 + 1x + x2 = 0 + 160000
1x + x2 = 0 + 160000

Combine like terms: 0 + 160000 = 160000
1x + x2 = 160000

The x term is 1x.  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 = 160000 + 0.25

Reorder the terms:
0.25 + x + x2 = 160000 + 0.25

Combine like terms: 160000 + 0.25 = 160000.25
0.25 + x + x2 = 160000.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 160000.25

Calculate the square root of the right side: 400.0003125

Break this problem into two subproblems by setting 
(x + 0.5) equal to 400.0003125 and -400.0003125.

Subproblem 1

x + 0.5 = 400.0003125 Simplifying x + 0.5 = 400.0003125 Reorder the terms: 0.5 + x = 400.0003125 Solving 0.5 + x = 400.0003125 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 = 400.0003125 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 400.0003125 + -0.5 x = 400.0003125 + -0.5 Combine like terms: 400.0003125 + -0.5 = 399.5003125 x = 399.5003125 Simplifying x = 399.5003125

Subproblem 2

x + 0.5 = -400.0003125 Simplifying x + 0.5 = -400.0003125 Reorder the terms: 0.5 + x = -400.0003125 Solving 0.5 + x = -400.0003125 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 = -400.0003125 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -400.0003125 + -0.5 x = -400.0003125 + -0.5 Combine like terms: -400.0003125 + -0.5 = -400.5003125 x = -400.5003125 Simplifying x = -400.5003125

Solution

The solution to the problem is based on the solutions from the subproblems. x = {399.5003125, -400.5003125}

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