5n^2-9=3n

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Solution for 5n^2-9=3n equation:


Simplifying
5n2 + -9 = 3n

Reorder the terms:
-9 + 5n2 = 3n

Solving
-9 + 5n2 = 3n

Solving for variable 'n'.

Reorder the terms:
-9 + -3n + 5n2 = 3n + -3n

Combine like terms: 3n + -3n = 0
-9 + -3n + 5n2 = 0

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

Divide each side by '5'.
-1.8 + -0.6n + n2 = 0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + -0.6n + 1.8 + n2 = 0 + 1.8

Reorder the terms:
-1.8 + 1.8 + -0.6n + n2 = 0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + -0.6n + n2 = 0 + 1.8
-0.6n + n2 = 0 + 1.8

Combine like terms: 0 + 1.8 = 1.8
-0.6n + n2 = 1.8

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 = 1.8 + 0.09

Reorder the terms:
0.09 + -0.6n + n2 = 1.8 + 0.09

Combine like terms: 1.8 + 0.09 = 1.89
0.09 + -0.6n + n2 = 1.89

Factor a perfect square on the left side:
(n + -0.3)(n + -0.3) = 1.89

Calculate the square root of the right side: 1.374772708

Break this problem into two subproblems by setting 
(n + -0.3) equal to 1.374772708 and -1.374772708.

Subproblem 1

n + -0.3 = 1.374772708 Simplifying n + -0.3 = 1.374772708 Reorder the terms: -0.3 + n = 1.374772708 Solving -0.3 + n = 1.374772708 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 = 1.374772708 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 1.374772708 + 0.3 n = 1.374772708 + 0.3 Combine like terms: 1.374772708 + 0.3 = 1.674772708 n = 1.674772708 Simplifying n = 1.674772708

Subproblem 2

n + -0.3 = -1.374772708 Simplifying n + -0.3 = -1.374772708 Reorder the terms: -0.3 + n = -1.374772708 Solving -0.3 + n = -1.374772708 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 = -1.374772708 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -1.374772708 + 0.3 n = -1.374772708 + 0.3 Combine like terms: -1.374772708 + 0.3 = -1.074772708 n = -1.074772708 Simplifying n = -1.074772708

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.674772708, -1.074772708}

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