10n^2-8n=1

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Solution for 10n^2-8n=1 equation:


Simplifying
10n2 + -8n = 1

Reorder the terms:
-8n + 10n2 = 1

Solving
-8n + 10n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -8n + 10n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -8n + 10n2 = 0

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

Divide each side by '10'.
-0.1 + -0.8n + n2 = 0

Move the constant term to the right:

Add '0.1' to each side of the equation.
-0.1 + -0.8n + 0.1 + n2 = 0 + 0.1

Reorder the terms:
-0.1 + 0.1 + -0.8n + n2 = 0 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + -0.8n + n2 = 0 + 0.1
-0.8n + n2 = 0 + 0.1

Combine like terms: 0 + 0.1 = 0.1
-0.8n + n2 = 0.1

The n term is -0.8n.  Take half its coefficient (-0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
-0.8n + 0.16 + n2 = 0.1 + 0.16

Reorder the terms:
0.16 + -0.8n + n2 = 0.1 + 0.16

Combine like terms: 0.1 + 0.16 = 0.26
0.16 + -0.8n + n2 = 0.26

Factor a perfect square on the left side:
(n + -0.4)(n + -0.4) = 0.26

Calculate the square root of the right side: 0.509901951

Break this problem into two subproblems by setting 
(n + -0.4) equal to 0.509901951 and -0.509901951.

Subproblem 1

n + -0.4 = 0.509901951 Simplifying n + -0.4 = 0.509901951 Reorder the terms: -0.4 + n = 0.509901951 Solving -0.4 + n = 0.509901951 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + n = 0.509901951 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = 0.509901951 + 0.4 n = 0.509901951 + 0.4 Combine like terms: 0.509901951 + 0.4 = 0.909901951 n = 0.909901951 Simplifying n = 0.909901951

Subproblem 2

n + -0.4 = -0.509901951 Simplifying n + -0.4 = -0.509901951 Reorder the terms: -0.4 + n = -0.509901951 Solving -0.4 + n = -0.509901951 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + n = -0.509901951 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = -0.509901951 + 0.4 n = -0.509901951 + 0.4 Combine like terms: -0.509901951 + 0.4 = -0.109901951 n = -0.109901951 Simplifying n = -0.109901951

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.909901951, -0.109901951}

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