5*n^2=9+n

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


Simplifying
5n2 = 9 + n

Solving
5n2 = 9 + n

Solving for variable 'n'.

Reorder the terms:
-9 + -1n + 5n2 = 9 + n + -9 + -1n

Reorder the terms:
-9 + -1n + 5n2 = 9 + -9 + n + -1n

Combine like terms: 9 + -9 = 0
-9 + -1n + 5n2 = 0 + n + -1n
-9 + -1n + 5n2 = n + -1n

Combine like terms: n + -1n = 0
-9 + -1n + 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 -0.2n.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2n + 0.01 + n2 = 1.8 + 0.01

Reorder the terms:
0.01 + -0.2n + n2 = 1.8 + 0.01

Combine like terms: 1.8 + 0.01 = 1.81
0.01 + -0.2n + n2 = 1.81

Factor a perfect square on the left side:
(n + -0.1)(n + -0.1) = 1.81

Calculate the square root of the right side: 1.345362405

Break this problem into two subproblems by setting 
(n + -0.1) equal to 1.345362405 and -1.345362405.

Subproblem 1

n + -0.1 = 1.345362405 Simplifying n + -0.1 = 1.345362405 Reorder the terms: -0.1 + n = 1.345362405 Solving -0.1 + n = 1.345362405 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + n = 1.345362405 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + n = 1.345362405 + 0.1 n = 1.345362405 + 0.1 Combine like terms: 1.345362405 + 0.1 = 1.445362405 n = 1.445362405 Simplifying n = 1.445362405

Subproblem 2

n + -0.1 = -1.345362405 Simplifying n + -0.1 = -1.345362405 Reorder the terms: -0.1 + n = -1.345362405 Solving -0.1 + n = -1.345362405 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + n = -1.345362405 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + n = -1.345362405 + 0.1 n = -1.345362405 + 0.1 Combine like terms: -1.345362405 + 0.1 = -1.245362405 n = -1.245362405 Simplifying n = -1.245362405

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.445362405, -1.245362405}

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