5=n(5n-9)

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Solution for 5=n(5n-9) equation:


Simplifying
5 = n(5n + -9)

Reorder the terms:
5 = n(-9 + 5n)
5 = (-9 * n + 5n * n)
5 = (-9n + 5n2)

Solving
5 = -9n + 5n2

Solving for variable 'n'.

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

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

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

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

Divide each side by '-5'.
-1 + -1.8n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1.8n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -1.8n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1.8n + n2 = 0 + 1
-1.8n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1.8n + n2 = 1

The n term is -1.8n.  Take half its coefficient (-0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
-1.8n + 0.81 + n2 = 1 + 0.81

Reorder the terms:
0.81 + -1.8n + n2 = 1 + 0.81

Combine like terms: 1 + 0.81 = 1.81
0.81 + -1.8n + n2 = 1.81

Factor a perfect square on the left side:
(n + -0.9)(n + -0.9) = 1.81

Calculate the square root of the right side: 1.345362405

Break this problem into two subproblems by setting 
(n + -0.9) equal to 1.345362405 and -1.345362405.

Subproblem 1

n + -0.9 = 1.345362405 Simplifying n + -0.9 = 1.345362405 Reorder the terms: -0.9 + n = 1.345362405 Solving -0.9 + n = 1.345362405 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + n = 1.345362405 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = 1.345362405 + 0.9 n = 1.345362405 + 0.9 Combine like terms: 1.345362405 + 0.9 = 2.245362405 n = 2.245362405 Simplifying n = 2.245362405

Subproblem 2

n + -0.9 = -1.345362405 Simplifying n + -0.9 = -1.345362405 Reorder the terms: -0.9 + n = -1.345362405 Solving -0.9 + n = -1.345362405 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + n = -1.345362405 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = -1.345362405 + 0.9 n = -1.345362405 + 0.9 Combine like terms: -1.345362405 + 0.9 = -0.445362405 n = -0.445362405 Simplifying n = -0.445362405

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.245362405, -0.445362405}

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