x(27-2x)=84

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Solution for x(27-2x)=84 equation: 


Simplifying
x(27 + -2x) = 84
(27 * x + -2x * x) = 84
(27x + -2x2) = 84

Solving
27x + -2x2 = 84

Solving for variable 'x'.

Reorder the terms:
-84 + 27x + -2x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 27x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
42 + -13.5x + x2 = 0

Move the constant term to the right:

Add '-42' to each side of the equation.
42 + -13.5x + -42 + x2 = 0 + -42

Reorder the terms:
42 + -42 + -13.5x + x2 = 0 + -42

Combine like terms: 42 + -42 = 0
0 + -13.5x + x2 = 0 + -42
-13.5x + x2 = 0 + -42

Combine like terms: 0 + -42 = -42
-13.5x + x2 = -42

The x term is -13.5x.  Take half its coefficient (-6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
-13.5x + 45.5625 + x2 = -42 + 45.5625

Reorder the terms:
45.5625 + -13.5x + x2 = -42 + 45.5625

Combine like terms: -42 + 45.5625 = 3.5625
45.5625 + -13.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + -6.75)(x + -6.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + -6.75) equal to 1.887458609 and -1.887458609.


Subproblem 1
x + -6.75 = 1.887458609

Simplifying
x + -6.75 = 1.887458609

Reorder the terms:
-6.75 + x = 1.887458609

Solving
-6.75 + x = 1.887458609

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = 1.887458609 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = 1.887458609 + 6.75
x = 1.887458609 + 6.75

Combine like terms: 1.887458609 + 6.75 = 8.637458609
x = 8.637458609

Simplifying
x = 8.637458609


Subproblem 2
x + -6.75 = -1.887458609

Simplifying
x + -6.75 = -1.887458609

Reorder the terms:
-6.75 + x = -1.887458609

Solving
-6.75 + x = -1.887458609

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = -1.887458609 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = -1.887458609 + 6.75
x = -1.887458609 + 6.75

Combine like terms: -1.887458609 + 6.75 = 4.862541391
x = 4.862541391

Simplifying
x = 4.862541391


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.637458609, 4.862541391}

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