3x^2-13x=1

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Solution for 3x^2-13x=1 equation:


Simplifying
3x2 + -13x = 1

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

Solving
-13x + 3x2 = 1

Solving for variable 'x'.

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

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

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

Divide each side by '3'.
-0.3333333333 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -4.333333333x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -4.333333333x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -4.333333333x + x2 = 0 + 0.3333333333
-4.333333333x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-4.333333333x + x2 = 0.3333333333

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = 0.3333333333 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 0.3333333333 + 4.694444446

Combine like terms: 0.3333333333 + 4.694444446 = 5.0277777793
4.694444446 + -4.333333333x + x2 = 5.0277777793

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 5.0277777793

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.242270675 and -2.242270675.

Subproblem 1

x + -2.166666667 = 2.242270675 Simplifying x + -2.166666667 = 2.242270675 Reorder the terms: -2.166666667 + x = 2.242270675 Solving -2.166666667 + x = 2.242270675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + x = 2.242270675 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + x = 2.242270675 + 2.166666667 x = 2.242270675 + 2.166666667 Combine like terms: 2.242270675 + 2.166666667 = 4.408937342 x = 4.408937342 Simplifying x = 4.408937342

Subproblem 2

x + -2.166666667 = -2.242270675 Simplifying x + -2.166666667 = -2.242270675 Reorder the terms: -2.166666667 + x = -2.242270675 Solving -2.166666667 + x = -2.242270675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + x = -2.242270675 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + x = -2.242270675 + 2.166666667 x = -2.242270675 + 2.166666667 Combine like terms: -2.242270675 + 2.166666667 = -0.075604008 x = -0.075604008 Simplifying x = -0.075604008

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.408937342, -0.075604008}

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