(10+x)(10-x)=x-6

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Solution for (10+x)(10-x)=x-6 equation:


Simplifying
(10 + x)(10 + -1x) = x + -6

Multiply (10 + x) * (10 + -1x)
(10(10 + -1x) + x(10 + -1x)) = x + -6
((10 * 10 + -1x * 10) + x(10 + -1x)) = x + -6
((100 + -10x) + x(10 + -1x)) = x + -6
(100 + -10x + (10 * x + -1x * x)) = x + -6
(100 + -10x + (10x + -1x2)) = x + -6

Combine like terms: -10x + 10x = 0
(100 + 0 + -1x2) = x + -6
(100 + -1x2) = x + -6

Reorder the terms:
100 + -1x2 = -6 + x

Solving
100 + -1x2 = -6 + x

Solving for variable 'x'.

Reorder the terms:
100 + 6 + -1x + -1x2 = -6 + x + 6 + -1x

Combine like terms: 100 + 6 = 106
106 + -1x + -1x2 = -6 + x + 6 + -1x

Reorder the terms:
106 + -1x + -1x2 = -6 + 6 + x + -1x

Combine like terms: -6 + 6 = 0
106 + -1x + -1x2 = 0 + x + -1x
106 + -1x + -1x2 = x + -1x

Combine like terms: x + -1x = 0
106 + -1x + -1x2 = 0

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

Divide each side by '-1'.
-106 + x + x2 = 0

Move the constant term to the right:

Add '106' to each side of the equation.
-106 + x + 106 + x2 = 0 + 106

Reorder the terms:
-106 + 106 + x + x2 = 0 + 106

Combine like terms: -106 + 106 = 0
0 + x + x2 = 0 + 106
x + x2 = 0 + 106

Combine like terms: 0 + 106 = 106
x + x2 = 106

The x term is x.  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.
 + 0.25 + x2 = 106 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 106 + 0.25

Combine like terms: 106 + 0.25 = 106.25
1.25 + x2 = 106.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 106.25

Calculate the square root of the right side: 10.307764064

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.307764064 and -10.307764064.

Subproblem 1

x + 0.5 = 10.307764064 Simplifying x + 0.5 = 10.307764064 Reorder the terms: 0.5 + x = 10.307764064 Solving 0.5 + x = 10.307764064 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 = 10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.307764064 + -0.5 x = 10.307764064 + -0.5 Combine like terms: 10.307764064 + -0.5 = 9.807764064 x = 9.807764064 Simplifying x = 9.807764064

Subproblem 2

x + 0.5 = -10.307764064 Simplifying x + 0.5 = -10.307764064 Reorder the terms: 0.5 + x = -10.307764064 Solving 0.5 + x = -10.307764064 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 = -10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.307764064 + -0.5 x = -10.307764064 + -0.5 Combine like terms: -10.307764064 + -0.5 = -10.807764064 x = -10.807764064 Simplifying x = -10.807764064

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.807764064, -10.807764064}

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