x(x+2)=3(x+4)+1

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Solution for x(x+2)=3(x+4)+1 equation:


Simplifying
x(x + 2) = 3(x + 4) + 1

Reorder the terms:
x(2 + x) = 3(x + 4) + 1
(2 * x + x * x) = 3(x + 4) + 1
(2x + x2) = 3(x + 4) + 1

Reorder the terms:
2x + x2 = 3(4 + x) + 1
2x + x2 = (4 * 3 + x * 3) + 1
2x + x2 = (12 + 3x) + 1

Reorder the terms:
2x + x2 = 12 + 1 + 3x

Combine like terms: 12 + 1 = 13
2x + x2 = 13 + 3x

Solving
2x + x2 = 13 + 3x

Solving for variable 'x'.

Reorder the terms:
-13 + 2x + -3x + x2 = 13 + 3x + -13 + -3x

Combine like terms: 2x + -3x = -1x
-13 + -1x + x2 = 13 + 3x + -13 + -3x

Reorder the terms:
-13 + -1x + x2 = 13 + -13 + 3x + -3x

Combine like terms: 13 + -13 = 0
-13 + -1x + x2 = 0 + 3x + -3x
-13 + -1x + x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-13 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -1x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + -1x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -1x + x2 = 0 + 13
-1x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
-1x + x2 = 13

The x term is -1x.  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.
-1x + 0.25 + x2 = 13 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 13 + 0.25

Combine like terms: 13 + 0.25 = 13.25
0.25 + -1x + x2 = 13.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(x + -0.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

x + -0.5 = 3.640054945 Simplifying x + -0.5 = 3.640054945 Reorder the terms: -0.5 + x = 3.640054945 Solving -0.5 + x = 3.640054945 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 = 3.640054945 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 3.640054945 + 0.5 x = 3.640054945 + 0.5 Combine like terms: 3.640054945 + 0.5 = 4.140054945 x = 4.140054945 Simplifying x = 4.140054945

Subproblem 2

x + -0.5 = -3.640054945 Simplifying x + -0.5 = -3.640054945 Reorder the terms: -0.5 + x = -3.640054945 Solving -0.5 + x = -3.640054945 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 = -3.640054945 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -3.640054945 + 0.5 x = -3.640054945 + 0.5 Combine like terms: -3.640054945 + 0.5 = -3.140054945 x = -3.140054945 Simplifying x = -3.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.140054945, -3.140054945}

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