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 = 0.45
0.25 + -1x + x2 = 0.45

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 0.45

Calculate the square root of the right side: 0.670820393

Break this problem into two subproblems by setting 
(x + -0.5) equal to 0.670820393 and -0.670820393.

Subproblem 1

x + -0.5 = 0.670820393 Simplifying x + -0.5 = 0.670820393 Reorder the terms: -0.5 + x = 0.670820393 Solving -0.5 + x = 0.670820393 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 = 0.670820393 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.670820393 + 0.5 x = 0.670820393 + 0.5 Combine like terms: 0.670820393 + 0.5 = 1.170820393 x = 1.170820393 Simplifying x = 1.170820393

Subproblem 2

x + -0.5 = -0.670820393 Simplifying x + -0.5 = -0.670820393 Reorder the terms: -0.5 + x = -0.670820393 Solving -0.5 + x = -0.670820393 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 = -0.670820393 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.670820393 + 0.5 x = -0.670820393 + 0.5 Combine like terms: -0.670820393 + 0.5 = -0.170820393 x = -0.170820393 Simplifying x = -0.170820393

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.170820393, -0.170820393}

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