5n^2=1+5n

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Solution for 5n^2=1+5n equation:


Simplifying
5n2 = 1 + 5n

Solving
5n2 = 1 + 5n

Solving for variable 'n'.

Reorder the terms:
-1 + -5n + 5n2 = 1 + 5n + -1 + -5n

Reorder the terms:
-1 + -5n + 5n2 = 1 + -1 + 5n + -5n

Combine like terms: 1 + -1 = 0
-1 + -5n + 5n2 = 0 + 5n + -5n
-1 + -5n + 5n2 = 5n + -5n

Combine like terms: 5n + -5n = 0
-1 + -5n + 5n2 = 0

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

Divide each side by '5'.
-0.2 + -1n + n2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -1n + 0.2 + n2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -1n + n2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -1n + n2 = 0 + 0.2
-1n + n2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-1n + n2 = 0.2

The n term is -1n.  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.
-1n + 0.25 + n2 = 0.2 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 0.2 + 0.25

Combine like terms: 0.2 + 0.25 = 0.45
0.25 + -1n + n2 = 0.45

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

Calculate the square root of the right side: 0.670820393

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

Subproblem 1

n + -0.5 = 0.670820393 Simplifying n + -0.5 = 0.670820393 Reorder the terms: -0.5 + n = 0.670820393 Solving -0.5 + n = 0.670820393 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 0.670820393 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 0.670820393 + 0.5 n = 0.670820393 + 0.5 Combine like terms: 0.670820393 + 0.5 = 1.170820393 n = 1.170820393 Simplifying n = 1.170820393

Subproblem 2

n + -0.5 = -0.670820393 Simplifying n + -0.5 = -0.670820393 Reorder the terms: -0.5 + n = -0.670820393 Solving -0.5 + n = -0.670820393 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -0.670820393 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -0.670820393 + 0.5 n = -0.670820393 + 0.5 Combine like terms: -0.670820393 + 0.5 = -0.170820393 n = -0.170820393 Simplifying n = -0.170820393

Solution

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

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