x(x-7)*(-2)=1

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Solution for x(x-7)*(-2)=1 equation: 


Simplifying
x(x + -7)(-2) = 1

Reorder the terms:
x(-7 + x)(-2) = 1

Reorder the terms for easier multiplication:
-2x(-7 + x) = 1
(-7 * -2x + x * -2x) = 1
(14x + -2x2) = 1

Solving
14x + -2x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 14x + -2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 14x + -2x2 = 0

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

Divide each side by '-2'.
0.5 + -7x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -7x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -7x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -7x + x2 = 0 + -0.5
-7x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-7x + x2 = -0.5

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -0.5 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -0.5 + 12.25

Combine like terms: -0.5 + 12.25 = 11.75
12.25 + -7x + x2 = 11.75

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 11.75

Calculate the square root of the right side: 3.4278273

Break this problem into two subproblems by setting 
(x + -3.5) equal to 3.4278273 and -3.4278273.


Subproblem 1
x + -3.5 = 3.4278273

Simplifying
x + -3.5 = 3.4278273

Reorder the terms:
-3.5 + x = 3.4278273

Solving
-3.5 + x = 3.4278273

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = 3.4278273 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 3.4278273 + 3.5
x = 3.4278273 + 3.5

Combine like terms: 3.4278273 + 3.5 = 6.9278273
x = 6.9278273

Simplifying
x = 6.9278273


Subproblem 2
x + -3.5 = -3.4278273

Simplifying
x + -3.5 = -3.4278273

Reorder the terms:
-3.5 + x = -3.4278273

Solving
-3.5 + x = -3.4278273

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = -3.4278273 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -3.4278273 + 3.5
x = -3.4278273 + 3.5

Combine like terms: -3.4278273 + 3.5 = 0.0721727
x = 0.0721727

Simplifying
x = 0.0721727


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.9278273, 0.0721727}

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