5n^2+n=9

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Solution for 5n^2+n=9 equation:


Simplifying
5n2 + n = 9

Reorder the terms:
n + 5n2 = 9

Solving
n + 5n2 = 9

Solving for variable 'n'.

Reorder the terms:
-9 + n + 5n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + n + 5n2 = 0

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

Divide each side by '5'.
-1.8 + 0.2n + n2 = 0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + 0.2n + 1.8 + n2 = 0 + 1.8

Reorder the terms:
-1.8 + 1.8 + 0.2n + n2 = 0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + 0.2n + n2 = 0 + 1.8
0.2n + n2 = 0 + 1.8

Combine like terms: 0 + 1.8 = 1.8
0.2n + n2 = 1.8

The n term is n.  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.
0.2n + 0.25 + n2 = 1.8 + 0.25

Reorder the terms:
0.25 + 0.2n + n2 = 1.8 + 0.25

Combine like terms: 1.8 + 0.25 = 2.05
0.25 + 0.2n + n2 = 2.05

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 2.05

Calculate the square root of the right side: 1.431782106

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.431782106 and -1.431782106.

Subproblem 1

n + 0.5 = 1.431782106 Simplifying n + 0.5 = 1.431782106 Reorder the terms: 0.5 + n = 1.431782106 Solving 0.5 + n = 1.431782106 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 = 1.431782106 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.431782106 + -0.5 n = 1.431782106 + -0.5 Combine like terms: 1.431782106 + -0.5 = 0.931782106 n = 0.931782106 Simplifying n = 0.931782106

Subproblem 2

n + 0.5 = -1.431782106 Simplifying n + 0.5 = -1.431782106 Reorder the terms: 0.5 + n = -1.431782106 Solving 0.5 + n = -1.431782106 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 = -1.431782106 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.431782106 + -0.5 n = -1.431782106 + -0.5 Combine like terms: -1.431782106 + -0.5 = -1.931782106 n = -1.931782106 Simplifying n = -1.931782106

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.931782106, -1.931782106}

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