6x^2-14x+3=0

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Solution for 6x^2-14x+3=0 equation:


Simplifying
6x2 + -14x + 3 = 0

Reorder the terms:
3 + -14x + 6x2 = 0

Solving
3 + -14x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
0.5 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.333333333x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.333333333x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.333333333x + x2 = 0 + -0.5
-2.333333333x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.333333333x + x2 = -0.5

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = -0.5 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = -0.5 + 1.361111112

Combine like terms: -0.5 + 1.361111112 = 0.861111112
1.361111112 + -2.333333333x + x2 = 0.861111112

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 0.861111112

Calculate the square root of the right side: 0.927960728

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 0.927960728 and -0.927960728.

Subproblem 1

x + -1.166666667 = 0.927960728 Simplifying x + -1.166666667 = 0.927960728 Reorder the terms: -1.166666667 + x = 0.927960728 Solving -1.166666667 + x = 0.927960728 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = 0.927960728 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 0.927960728 + 1.166666667 x = 0.927960728 + 1.166666667 Combine like terms: 0.927960728 + 1.166666667 = 2.094627395 x = 2.094627395 Simplifying x = 2.094627395

Subproblem 2

x + -1.166666667 = -0.927960728 Simplifying x + -1.166666667 = -0.927960728 Reorder the terms: -1.166666667 + x = -0.927960728 Solving -1.166666667 + x = -0.927960728 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = -0.927960728 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -0.927960728 + 1.166666667 x = -0.927960728 + 1.166666667 Combine like terms: -0.927960728 + 1.166666667 = 0.238705939 x = 0.238705939 Simplifying x = 0.238705939

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.094627395, 0.238705939}

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