x^2+4x^2+20x+25=x^2+12x+36

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Solution for x^2+4x^2+20x+25=x^2+12x+36 equation:


Simplifying
x2 + 4x2 + 20x + 25 = x2 + 12x + 36

Reorder the terms:
25 + 20x + x2 + 4x2 = x2 + 12x + 36

Combine like terms: x2 + 4x2 = 5x2
25 + 20x + 5x2 = x2 + 12x + 36

Reorder the terms:
25 + 20x + 5x2 = 36 + 12x + x2

Solving
25 + 20x + 5x2 = 36 + 12x + x2

Solving for variable 'x'.

Reorder the terms:
25 + -36 + 20x + -12x + 5x2 + -1x2 = 36 + 12x + x2 + -36 + -12x + -1x2

Combine like terms: 25 + -36 = -11
-11 + 20x + -12x + 5x2 + -1x2 = 36 + 12x + x2 + -36 + -12x + -1x2

Combine like terms: 20x + -12x = 8x
-11 + 8x + 5x2 + -1x2 = 36 + 12x + x2 + -36 + -12x + -1x2

Combine like terms: 5x2 + -1x2 = 4x2
-11 + 8x + 4x2 = 36 + 12x + x2 + -36 + -12x + -1x2

Reorder the terms:
-11 + 8x + 4x2 = 36 + -36 + 12x + -12x + x2 + -1x2

Combine like terms: 36 + -36 = 0
-11 + 8x + 4x2 = 0 + 12x + -12x + x2 + -1x2
-11 + 8x + 4x2 = 12x + -12x + x2 + -1x2

Combine like terms: 12x + -12x = 0
-11 + 8x + 4x2 = 0 + x2 + -1x2
-11 + 8x + 4x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-11 + 8x + 4x2 = 0

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

Divide each side by '4'.
-2.75 + 2x + x2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + 2x + 2.75 + x2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + 2x + x2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + 2x + x2 = 0 + 2.75
2x + x2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
2x + x2 = 2.75

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 2.75 + 1

Reorder the terms:
1 + 2x + x2 = 2.75 + 1

Combine like terms: 2.75 + 1 = 3.75
1 + 2x + x2 = 3.75

Factor a perfect square on the left side:
(x + 1)(x + 1) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(x + 1) equal to 1.936491673 and -1.936491673.

Subproblem 1

x + 1 = 1.936491673 Simplifying x + 1 = 1.936491673 Reorder the terms: 1 + x = 1.936491673 Solving 1 + x = 1.936491673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 1.936491673 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.936491673 + -1 x = 1.936491673 + -1 Combine like terms: 1.936491673 + -1 = 0.936491673 x = 0.936491673 Simplifying x = 0.936491673

Subproblem 2

x + 1 = -1.936491673 Simplifying x + 1 = -1.936491673 Reorder the terms: 1 + x = -1.936491673 Solving 1 + x = -1.936491673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -1.936491673 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.936491673 + -1 x = -1.936491673 + -1 Combine like terms: -1.936491673 + -1 = -2.936491673 x = -2.936491673 Simplifying x = -2.936491673

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.936491673, -2.936491673}

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