5n^2=9n-2

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


Simplifying
5n2 = 9n + -2

Reorder the terms:
5n2 = -2 + 9n

Solving
5n2 = -2 + 9n

Solving for variable 'n'.

Reorder the terms:
2 + -9n + 5n2 = -2 + 9n + 2 + -9n

Reorder the terms:
2 + -9n + 5n2 = -2 + 2 + 9n + -9n

Combine like terms: -2 + 2 = 0
2 + -9n + 5n2 = 0 + 9n + -9n
2 + -9n + 5n2 = 9n + -9n

Combine like terms: 9n + -9n = 0
2 + -9n + 5n2 = 0

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

Divide each side by '5'.
0.4 + -1.8n + n2 = 0

Move the constant term to the right:

Add '-0.4' to each side of the equation.
0.4 + -1.8n + -0.4 + n2 = 0 + -0.4

Reorder the terms:
0.4 + -0.4 + -1.8n + n2 = 0 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + -1.8n + n2 = 0 + -0.4
-1.8n + n2 = 0 + -0.4

Combine like terms: 0 + -0.4 = -0.4
-1.8n + n2 = -0.4

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 = -0.4 + 0.81

Reorder the terms:
0.81 + -1.8n + n2 = -0.4 + 0.81

Combine like terms: -0.4 + 0.81 = 0.41
0.81 + -1.8n + n2 = 0.41

Factor a perfect square on the left side:
(n + -0.9)(n + -0.9) = 0.41

Calculate the square root of the right side: 0.640312424

Break this problem into two subproblems by setting 
(n + -0.9) equal to 0.640312424 and -0.640312424.

Subproblem 1

n + -0.9 = 0.640312424 Simplifying n + -0.9 = 0.640312424 Reorder the terms: -0.9 + n = 0.640312424 Solving -0.9 + n = 0.640312424 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 = 0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = 0.640312424 + 0.9 n = 0.640312424 + 0.9 Combine like terms: 0.640312424 + 0.9 = 1.540312424 n = 1.540312424 Simplifying n = 1.540312424

Subproblem 2

n + -0.9 = -0.640312424 Simplifying n + -0.9 = -0.640312424 Reorder the terms: -0.9 + n = -0.640312424 Solving -0.9 + n = -0.640312424 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 = -0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = -0.640312424 + 0.9 n = -0.640312424 + 0.9 Combine like terms: -0.640312424 + 0.9 = 0.259687576 n = 0.259687576 Simplifying n = 0.259687576

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.540312424, 0.259687576}

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