1000x=(100-x)(10-x)

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Solution for 1000x=(100-x)(10-x) equation:


Simplifying
1000x = (100 + -1x)(10 + -1x)

Multiply (100 + -1x) * (10 + -1x)
1000x = (100(10 + -1x) + -1x * (10 + -1x))
1000x = ((10 * 100 + -1x * 100) + -1x * (10 + -1x))
1000x = ((1000 + -100x) + -1x * (10 + -1x))
1000x = (1000 + -100x + (10 * -1x + -1x * -1x))
1000x = (1000 + -100x + (-10x + 1x2))

Combine like terms: -100x + -10x = -110x
1000x = (1000 + -110x + 1x2)

Solving
1000x = 1000 + -110x + 1x2

Solving for variable 'x'.

Reorder the terms:
-1000 + 1000x + 110x + -1x2 = 1000 + -110x + 1x2 + -1000 + 110x + -1x2

Combine like terms: 1000x + 110x = 1110x
-1000 + 1110x + -1x2 = 1000 + -110x + 1x2 + -1000 + 110x + -1x2

Reorder the terms:
-1000 + 1110x + -1x2 = 1000 + -1000 + -110x + 110x + 1x2 + -1x2

Combine like terms: 1000 + -1000 = 0
-1000 + 1110x + -1x2 = 0 + -110x + 110x + 1x2 + -1x2
-1000 + 1110x + -1x2 = -110x + 110x + 1x2 + -1x2

Combine like terms: -110x + 110x = 0
-1000 + 1110x + -1x2 = 0 + 1x2 + -1x2
-1000 + 1110x + -1x2 = 1x2 + -1x2

Combine like terms: 1x2 + -1x2 = 0
-1000 + 1110x + -1x2 = 0

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

Divide each side by '-1'.
1000 + -1110x + x2 = 0

Move the constant term to the right:

Add '-1000' to each side of the equation.
1000 + -1110x + -1000 + x2 = 0 + -1000

Reorder the terms:
1000 + -1000 + -1110x + x2 = 0 + -1000

Combine like terms: 1000 + -1000 = 0
0 + -1110x + x2 = 0 + -1000
-1110x + x2 = 0 + -1000

Combine like terms: 0 + -1000 = -1000
-1110x + x2 = -1000

The x term is -1110x.  Take half its coefficient (-555).
Square it (308025) and add it to both sides.

Add '308025' to each side of the equation.
-1110x + 308025 + x2 = -1000 + 308025

Reorder the terms:
308025 + -1110x + x2 = -1000 + 308025

Combine like terms: -1000 + 308025 = 307025
308025 + -1110x + x2 = 307025

Factor a perfect square on the left side:
(x + -555)(x + -555) = 307025

Calculate the square root of the right side: 554.098366718

Break this problem into two subproblems by setting 
(x + -555) equal to 554.098366718 and -554.098366718.

Subproblem 1

x + -555 = 554.098366718 Simplifying x + -555 = 554.098366718 Reorder the terms: -555 + x = 554.098366718 Solving -555 + x = 554.098366718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '555' to each side of the equation. -555 + 555 + x = 554.098366718 + 555 Combine like terms: -555 + 555 = 0 0 + x = 554.098366718 + 555 x = 554.098366718 + 555 Combine like terms: 554.098366718 + 555 = 1109.098366718 x = 1109.098366718 Simplifying x = 1109.098366718

Subproblem 2

x + -555 = -554.098366718 Simplifying x + -555 = -554.098366718 Reorder the terms: -555 + x = -554.098366718 Solving -555 + x = -554.098366718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '555' to each side of the equation. -555 + 555 + x = -554.098366718 + 555 Combine like terms: -555 + 555 = 0 0 + x = -554.098366718 + 555 x = -554.098366718 + 555 Combine like terms: -554.098366718 + 555 = 0.901633282 x = 0.901633282 Simplifying x = 0.901633282

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1109.098366718, 0.901633282}

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