6x^2-14x+5=0

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Solution for 6x^2-14x+5=0 equation:


Simplifying
6x2 + -14x + 5 = 0

Reorder the terms:
5 + -14x + 6x2 = 0

Solving
5 + -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'.
0.8333333333 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + -2.333333333x + -0.8333333333 + x2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + -2.333333333x + x2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + -2.333333333x + x2 = 0 + -0.8333333333
-2.333333333x + x2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
-2.333333333x + x2 = -0.8333333333

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 = -0.8333333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = -0.8333333333 + 1.361111112

Combine like terms: -0.8333333333 + 1.361111112 = 0.5277777787
1.361111112 + -2.333333333x + x2 = 0.5277777787

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 0.5277777787

Calculate the square root of the right side: 0.726483158

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 0.726483158 and -0.726483158.

Subproblem 1

x + -1.166666667 = 0.726483158 Simplifying x + -1.166666667 = 0.726483158 Reorder the terms: -1.166666667 + x = 0.726483158 Solving -1.166666667 + x = 0.726483158 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 = 0.726483158 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 0.726483158 + 1.166666667 x = 0.726483158 + 1.166666667 Combine like terms: 0.726483158 + 1.166666667 = 1.893149825 x = 1.893149825 Simplifying x = 1.893149825

Subproblem 2

x + -1.166666667 = -0.726483158 Simplifying x + -1.166666667 = -0.726483158 Reorder the terms: -1.166666667 + x = -0.726483158 Solving -1.166666667 + x = -0.726483158 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 = -0.726483158 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -0.726483158 + 1.166666667 x = -0.726483158 + 1.166666667 Combine like terms: -0.726483158 + 1.166666667 = 0.440183509 x = 0.440183509 Simplifying x = 0.440183509

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.893149825, 0.440183509}

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