5n^2-6=30n

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Solution for 5n^2-6=30n equation:


Simplifying
5n2 + -6 = 30n

Reorder the terms:
-6 + 5n2 = 30n

Solving
-6 + 5n2 = 30n

Solving for variable 'n'.

Reorder the terms:
-6 + -30n + 5n2 = 30n + -30n

Combine like terms: 30n + -30n = 0
-6 + -30n + 5n2 = 0

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

Divide each side by '5'.
-1.2 + -6n + n2 = 0

Move the constant term to the right:

Add '1.2' to each side of the equation.
-1.2 + -6n + 1.2 + n2 = 0 + 1.2

Reorder the terms:
-1.2 + 1.2 + -6n + n2 = 0 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + -6n + n2 = 0 + 1.2
-6n + n2 = 0 + 1.2

Combine like terms: 0 + 1.2 = 1.2
-6n + n2 = 1.2

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = 1.2 + 9

Reorder the terms:
9 + -6n + n2 = 1.2 + 9

Combine like terms: 1.2 + 9 = 10.2
9 + -6n + n2 = 10.2

Factor a perfect square on the left side:
(n + -3)(n + -3) = 10.2

Calculate the square root of the right side: 3.193743885

Break this problem into two subproblems by setting 
(n + -3) equal to 3.193743885 and -3.193743885.

Subproblem 1

n + -3 = 3.193743885 Simplifying n + -3 = 3.193743885 Reorder the terms: -3 + n = 3.193743885 Solving -3 + n = 3.193743885 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 3.193743885 + 3 Combine like terms: -3 + 3 = 0 0 + n = 3.193743885 + 3 n = 3.193743885 + 3 Combine like terms: 3.193743885 + 3 = 6.193743885 n = 6.193743885 Simplifying n = 6.193743885

Subproblem 2

n + -3 = -3.193743885 Simplifying n + -3 = -3.193743885 Reorder the terms: -3 + n = -3.193743885 Solving -3 + n = -3.193743885 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -3.193743885 + 3 Combine like terms: -3 + 3 = 0 0 + n = -3.193743885 + 3 n = -3.193743885 + 3 Combine like terms: -3.193743885 + 3 = -0.193743885 n = -0.193743885 Simplifying n = -0.193743885

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.193743885, -0.193743885}

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