5a^2-5a+1=0

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Solution for 5a^2-5a+1=0 equation:


Simplifying
5a2 + -5a + 1 = 0

Reorder the terms:
1 + -5a + 5a2 = 0

Solving
1 + -5a + 5a2 = 0

Solving for variable 'a'.

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

Divide each side by '5'.
0.2 + -1a + a2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -1a + -0.2 + a2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -1a + a2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -1a + a2 = 0 + -0.2
-1a + a2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-1a + a2 = -0.2

The a term is -1a.  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.
-1a + 0.25 + a2 = -0.2 + 0.25

Reorder the terms:
0.25 + -1a + a2 = -0.2 + 0.25

Combine like terms: -0.2 + 0.25 = 0.05
0.25 + -1a + a2 = 0.05

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

Calculate the square root of the right side: 0.223606798

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

Subproblem 1

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

Subproblem 2

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

Solution

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

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