5n^2-3n=1

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Solution for 5n^2-3n=1 equation:


Simplifying
5n2 + -3n = 1

Reorder the terms:
-3n + 5n2 = 1

Solving
-3n + 5n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -3n + 5n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -3n + 5n2 = 0

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

Divide each side by '5'.
-0.2 + -0.6n + n2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -0.6n + 0.2 + n2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -0.6n + n2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -0.6n + n2 = 0 + 0.2
-0.6n + n2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-0.6n + n2 = 0.2

The n term is -0.6n.  Take half its coefficient (-0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
-0.6n + 0.09 + n2 = 0.2 + 0.09

Reorder the terms:
0.09 + -0.6n + n2 = 0.2 + 0.09

Combine like terms: 0.2 + 0.09 = 0.29
0.09 + -0.6n + n2 = 0.29

Factor a perfect square on the left side:
(n + -0.3)(n + -0.3) = 0.29

Calculate the square root of the right side: 0.538516481

Break this problem into two subproblems by setting 
(n + -0.3) equal to 0.538516481 and -0.538516481.

Subproblem 1

n + -0.3 = 0.538516481 Simplifying n + -0.3 = 0.538516481 Reorder the terms: -0.3 + n = 0.538516481 Solving -0.3 + n = 0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = 0.538516481 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 0.538516481 + 0.3 n = 0.538516481 + 0.3 Combine like terms: 0.538516481 + 0.3 = 0.838516481 n = 0.838516481 Simplifying n = 0.838516481

Subproblem 2

n + -0.3 = -0.538516481 Simplifying n + -0.3 = -0.538516481 Reorder the terms: -0.3 + n = -0.538516481 Solving -0.3 + n = -0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = -0.538516481 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -0.538516481 + 0.3 n = -0.538516481 + 0.3 Combine like terms: -0.538516481 + 0.3 = -0.238516481 n = -0.238516481 Simplifying n = -0.238516481

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.838516481, -0.238516481}

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