5x^2=5x-1

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Solution for 5x^2=5x-1 equation:


Simplifying
5x2 = 5x + -1

Reorder the terms:
5x2 = -1 + 5x

Solving
5x2 = -1 + 5x

Solving for variable 'x'.

Reorder the terms:
1 + -5x + 5x2 = -1 + 5x + 1 + -5x

Reorder the terms:
1 + -5x + 5x2 = -1 + 1 + 5x + -5x

Combine like terms: -1 + 1 = 0
1 + -5x + 5x2 = 0 + 5x + -5x
1 + -5x + 5x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
1 + -5x + 5x2 = 0

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

Divide each side by '5'.
0.2 + -1x + x2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -1x + -0.2 + x2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -1x + x2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -1x + x2 = 0 + -0.2
-1x + x2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-1x + x2 = -0.2

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = -0.2 + 0.25

Reorder the terms:
0.25 + -1x + x2 = -0.2 + 0.25

Combine like terms: -0.2 + 0.25 = 0.05
0.25 + -1x + x2 = 0.05

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 0.05

Calculate the square root of the right side: 0.223606798

Break this problem into two subproblems by setting 
(x + -0.5) equal to 0.223606798 and -0.223606798.

Subproblem 1

x + -0.5 = 0.223606798 Simplifying x + -0.5 = 0.223606798 Reorder the terms: -0.5 + x = 0.223606798 Solving -0.5 + x = 0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.223606798 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.223606798 + 0.5 x = 0.223606798 + 0.5 Combine like terms: 0.223606798 + 0.5 = 0.723606798 x = 0.723606798 Simplifying x = 0.723606798

Subproblem 2

x + -0.5 = -0.223606798 Simplifying x + -0.5 = -0.223606798 Reorder the terms: -0.5 + x = -0.223606798 Solving -0.5 + x = -0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.223606798 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.223606798 + 0.5 x = -0.223606798 + 0.5 Combine like terms: -0.223606798 + 0.5 = 0.276393202 x = 0.276393202 Simplifying x = 0.276393202

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.723606798, 0.276393202}

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