10x-6x^2=1

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


Simplifying
10x + -6x2 = 1

Solving
10x + -6x2 = 1

Solving for variable 'x'.

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

Combine like terms: 1 + -1 = 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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