3x^2-13x+3=0

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Solution for 3x^2-13x+3=0 equation:


Simplifying
3x2 + -13x + 3 = 0

Reorder the terms:
3 + -13x + 3x2 = 0

Solving
3 + -13x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
1 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4.333333333x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -4.333333333x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4.333333333x + x2 = 0 + -1
-4.333333333x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4.333333333x + x2 = -1

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 = -1 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = -1 + 4.694444446

Combine like terms: -1 + 4.694444446 = 3.694444446
4.694444446 + -4.333333333x + x2 = 3.694444446

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 3.694444446

Calculate the square root of the right side: 1.922093766

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 1.922093766 and -1.922093766.

Subproblem 1

x + -2.166666667 = 1.922093766 Simplifying x + -2.166666667 = 1.922093766 Reorder the terms: -2.166666667 + x = 1.922093766 Solving -2.166666667 + x = 1.922093766 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 = 1.922093766 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + x = 1.922093766 + 2.166666667 x = 1.922093766 + 2.166666667 Combine like terms: 1.922093766 + 2.166666667 = 4.088760433 x = 4.088760433 Simplifying x = 4.088760433

Subproblem 2

x + -2.166666667 = -1.922093766 Simplifying x + -2.166666667 = -1.922093766 Reorder the terms: -2.166666667 + x = -1.922093766 Solving -2.166666667 + x = -1.922093766 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 = -1.922093766 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + x = -1.922093766 + 2.166666667 x = -1.922093766 + 2.166666667 Combine like terms: -1.922093766 + 2.166666667 = 0.244572901 x = 0.244572901 Simplifying x = 0.244572901

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.088760433, 0.244572901}

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