3x+13=x(x+6)+1

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Solution for 3x+13=x(x+6)+1 equation:


Simplifying
3x + 13 = x(x + 6) + 1

Reorder the terms:
13 + 3x = x(x + 6) + 1

Reorder the terms:
13 + 3x = x(6 + x) + 1
13 + 3x = (6 * x + x * x) + 1
13 + 3x = (6x + x2) + 1

Reorder the terms:
13 + 3x = 1 + 6x + x2

Solving
13 + 3x = 1 + 6x + x2

Solving for variable 'x'.

Reorder the terms:
13 + -1 + 3x + -6x + -1x2 = 1 + 6x + x2 + -1 + -6x + -1x2

Combine like terms: 13 + -1 = 12
12 + 3x + -6x + -1x2 = 1 + 6x + x2 + -1 + -6x + -1x2

Combine like terms: 3x + -6x = -3x
12 + -3x + -1x2 = 1 + 6x + x2 + -1 + -6x + -1x2

Reorder the terms:
12 + -3x + -1x2 = 1 + -1 + 6x + -6x + x2 + -1x2

Combine like terms: 1 + -1 = 0
12 + -3x + -1x2 = 0 + 6x + -6x + x2 + -1x2
12 + -3x + -1x2 = 6x + -6x + x2 + -1x2

Combine like terms: 6x + -6x = 0
12 + -3x + -1x2 = 0 + x2 + -1x2
12 + -3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
12 + -3x + -1x2 = 0

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

Divide each side by '-1'.
-12 + 3x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 3x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + 3x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 3x + x2 = 0 + 12
3x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
3x + x2 = 12

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 12 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 12 + 2.25

Combine like terms: 12 + 2.25 = 14.25
2.25 + 3x + x2 = 14.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(x + 1.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

x + 1.5 = 3.774917218 Simplifying x + 1.5 = 3.774917218 Reorder the terms: 1.5 + x = 3.774917218 Solving 1.5 + x = 3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 3.774917218 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 3.774917218 + -1.5 x = 3.774917218 + -1.5 Combine like terms: 3.774917218 + -1.5 = 2.274917218 x = 2.274917218 Simplifying x = 2.274917218

Subproblem 2

x + 1.5 = -3.774917218 Simplifying x + 1.5 = -3.774917218 Reorder the terms: 1.5 + x = -3.774917218 Solving 1.5 + x = -3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -3.774917218 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -3.774917218 + -1.5 x = -3.774917218 + -1.5 Combine like terms: -3.774917218 + -1.5 = -5.274917218 x = -5.274917218 Simplifying x = -5.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.274917218, -5.274917218}

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