6x^2=10x-1

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Solution for 6x^2=10x-1 equation:


Simplifying
6x2 = 10x + -1

Reorder the terms:
6x2 = -1 + 10x

Solving
6x2 = -1 + 10x

Solving for variable 'x'.

Reorder the terms:
1 + -10x + 6x2 = -1 + 10x + 1 + -10x

Reorder the terms:
1 + -10x + 6x2 = -1 + 1 + 10x + -10x

Combine like terms: -1 + 1 = 0
1 + -10x + 6x2 = 0 + 10x + -10x
1 + -10x + 6x2 = 10x + -10x

Combine like terms: 10x + -10x = 0
1 + -10x + 6x2 = 0

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

Divide each side by '6'.
0.1666666667 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -1.666666667x + -0.1666666667 + x2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + -1.666666667x + x2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + -1.666666667x + x2 = 0 + -0.1666666667
-1.666666667x + x2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
-1.666666667x + x2 = -0.1666666667

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = -0.1666666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = -0.1666666667 + 0.6944444447

Combine like terms: -0.1666666667 + 0.6944444447 = 0.527777778
0.6944444447 + -1.666666667x + x2 = 0.527777778

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 0.527777778

Calculate the square root of the right side: 0.726483157

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 0.726483157 and -0.726483157.

Subproblem 1

x + -0.8333333335 = 0.726483157 Simplifying x + -0.8333333335 = 0.726483157 Reorder the terms: -0.8333333335 + x = 0.726483157 Solving -0.8333333335 + x = 0.726483157 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 0.726483157 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 0.726483157 + 0.8333333335 x = 0.726483157 + 0.8333333335 Combine like terms: 0.726483157 + 0.8333333335 = 1.5598164905 x = 1.5598164905 Simplifying x = 1.5598164905

Subproblem 2

x + -0.8333333335 = -0.726483157 Simplifying x + -0.8333333335 = -0.726483157 Reorder the terms: -0.8333333335 + x = -0.726483157 Solving -0.8333333335 + x = -0.726483157 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -0.726483157 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -0.726483157 + 0.8333333335 x = -0.726483157 + 0.8333333335 Combine like terms: -0.726483157 + 0.8333333335 = 0.1068501765 x = 0.1068501765 Simplifying x = 0.1068501765

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.5598164905, 0.1068501765}

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