100=x^2+4x^2-40x-100

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Solution for 100=x^2+4x^2-40x-100 equation: 


Simplifying
100 = x2 + 4x2 + -40x + -100

Reorder the terms:
100 = -100 + -40x + x2 + 4x2

Combine like terms: x2 + 4x2 = 5x2
100 = -100 + -40x + 5x2

Solving
100 = -100 + -40x + 5x2

Solving for variable 'x'.

Combine like terms: 100 + 100 = 200
200 + 40x + -5x2 = -100 + -40x + 5x2 + 100 + 40x + -5x2

Reorder the terms:
200 + 40x + -5x2 = -100 + 100 + -40x + 40x + 5x2 + -5x2

Combine like terms: -100 + 100 = 0
200 + 40x + -5x2 = 0 + -40x + 40x + 5x2 + -5x2
200 + 40x + -5x2 = -40x + 40x + 5x2 + -5x2

Combine like terms: -40x + 40x = 0
200 + 40x + -5x2 = 0 + 5x2 + -5x2
200 + 40x + -5x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
200 + 40x + -5x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(40 + 8x + -1x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(40 + 8x + -1x2)' equal to zero and attempt to solve:

Simplifying
40 + 8x + -1x2 = 0

Solving
40 + 8x + -1x2 = 0

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

Divide each side by '-1'.
-40 + -8x + x2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -8x + 40 + x2 = 0 + 40

Reorder the terms:
-40 + 40 + -8x + x2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -8x + x2 = 0 + 40
-8x + x2 = 0 + 40

Combine like terms: 0 + 40 = 40
-8x + x2 = 40

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = 40 + 16

Reorder the terms:
16 + -8x + x2 = 40 + 16

Combine like terms: 40 + 16 = 56
16 + -8x + x2 = 56

Factor a perfect square on the left side:
(x + -4)(x + -4) = 56

Calculate the square root of the right side: 7.483314774

Break this problem into two subproblems by setting 
(x + -4) equal to 7.483314774 and -7.483314774.


Subproblem 1
x + -4 = 7.483314774

Simplifying
x + -4 = 7.483314774

Reorder the terms:
-4 + x = 7.483314774

Solving
-4 + x = 7.483314774

Solving for variable 'x'.

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

Add '4' to each side of the equation.
-4 + 4 + x = 7.483314774 + 4

Combine like terms: -4 + 4 = 0
0 + x = 7.483314774 + 4
x = 7.483314774 + 4

Combine like terms: 7.483314774 + 4 = 11.483314774
x = 11.483314774

Simplifying
x = 11.483314774


Subproblem 2
x + -4 = -7.483314774

Simplifying
x + -4 = -7.483314774

Reorder the terms:
-4 + x = -7.483314774

Solving
-4 + x = -7.483314774

Solving for variable 'x'.

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

Add '4' to each side of the equation.
-4 + 4 + x = -7.483314774 + 4

Combine like terms: -4 + 4 = 0
0 + x = -7.483314774 + 4
x = -7.483314774 + 4

Combine like terms: -7.483314774 + 4 = -3.483314774
x = -3.483314774

Simplifying
x = -3.483314774


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.483314774, -3.483314774}
Solution
x = {11.483314774, -3.483314774}

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