x=3x(90-x)+18

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Solution for x=3x(90-x)+18 equation:


Simplifying
x = 3x(90 + -1x) + 18
x = (90 * 3x + -1x * 3x) + 18
x = (270x + -3x2) + 18

Reorder the terms:
x = 18 + 270x + -3x2

Solving
x = 18 + 270x + -3x2

Solving for variable 'x'.

Reorder the terms:
-18 + x + -270x + 3x2 = 18 + 270x + -3x2 + -18 + -270x + 3x2

Combine like terms: x + -270x = -269x
-18 + -269x + 3x2 = 18 + 270x + -3x2 + -18 + -270x + 3x2

Reorder the terms:
-18 + -269x + 3x2 = 18 + -18 + 270x + -270x + -3x2 + 3x2

Combine like terms: 18 + -18 = 0
-18 + -269x + 3x2 = 0 + 270x + -270x + -3x2 + 3x2
-18 + -269x + 3x2 = 270x + -270x + -3x2 + 3x2

Combine like terms: 270x + -270x = 0
-18 + -269x + 3x2 = 0 + -3x2 + 3x2
-18 + -269x + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-18 + -269x + 3x2 = 0

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

Divide each side by '3'.
-6 + -89.66666667x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -89.66666667x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + -89.66666667x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -89.66666667x + x2 = 0 + 6
-89.66666667x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
-89.66666667x + x2 = 6

The x term is -89.66666667x.  Take half its coefficient (-44.83333334).
Square it (2010.027778) and add it to both sides.

Add '2010.027778' to each side of the equation.
-89.66666667x + 2010.027778 + x2 = 6 + 2010.027778

Reorder the terms:
2010.027778 + -89.66666667x + x2 = 6 + 2010.027778

Combine like terms: 6 + 2010.027778 = 2016.027778
2010.027778 + -89.66666667x + x2 = 2016.027778

Factor a perfect square on the left side:
(x + -44.83333334)(x + -44.83333334) = 2016.027778

Calculate the square root of the right side: 44.900197973

Break this problem into two subproblems by setting 
(x + -44.83333334) equal to 44.900197973 and -44.900197973.

Subproblem 1

x + -44.83333334 = 44.900197973 Simplifying x + -44.83333334 = 44.900197973 Reorder the terms: -44.83333334 + x = 44.900197973 Solving -44.83333334 + x = 44.900197973 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '44.83333334' to each side of the equation. -44.83333334 + 44.83333334 + x = 44.900197973 + 44.83333334 Combine like terms: -44.83333334 + 44.83333334 = 0.00000000 0.00000000 + x = 44.900197973 + 44.83333334 x = 44.900197973 + 44.83333334 Combine like terms: 44.900197973 + 44.83333334 = 89.733531313 x = 89.733531313 Simplifying x = 89.733531313

Subproblem 2

x + -44.83333334 = -44.900197973 Simplifying x + -44.83333334 = -44.900197973 Reorder the terms: -44.83333334 + x = -44.900197973 Solving -44.83333334 + x = -44.900197973 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '44.83333334' to each side of the equation. -44.83333334 + 44.83333334 + x = -44.900197973 + 44.83333334 Combine like terms: -44.83333334 + 44.83333334 = 0.00000000 0.00000000 + x = -44.900197973 + 44.83333334 x = -44.900197973 + 44.83333334 Combine like terms: -44.900197973 + 44.83333334 = -0.066864633 x = -0.066864633 Simplifying x = -0.066864633

Solution

The solution to the problem is based on the solutions from the subproblems. x = {89.733531313, -0.066864633}

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