5n^2+5n=6

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Solution for 5n^2+5n=6 equation:


Simplifying
5n2 + 5n = 6

Reorder the terms:
5n + 5n2 = 6

Solving
5n + 5n2 = 6

Solving for variable 'n'.

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

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

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

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

Move the constant term to the right:

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

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

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

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

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + n2 = 1.2 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 1.2 + 0.25

Combine like terms: 1.2 + 0.25 = 1.45
1.25 + n2 = 1.45

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 1.45

Calculate the square root of the right side: 1.204159458

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.204159458 and -1.204159458.

Subproblem 1

n + 0.5 = 1.204159458 Simplifying n + 0.5 = 1.204159458 Reorder the terms: 0.5 + n = 1.204159458 Solving 0.5 + n = 1.204159458 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.204159458 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.204159458 + -0.5 n = 1.204159458 + -0.5 Combine like terms: 1.204159458 + -0.5 = 0.704159458 n = 0.704159458 Simplifying n = 0.704159458

Subproblem 2

n + 0.5 = -1.204159458 Simplifying n + 0.5 = -1.204159458 Reorder the terms: 0.5 + n = -1.204159458 Solving 0.5 + n = -1.204159458 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.204159458 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.204159458 + -0.5 n = -1.204159458 + -0.5 Combine like terms: -1.204159458 + -0.5 = -1.704159458 n = -1.704159458 Simplifying n = -1.704159458

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.704159458, -1.704159458}

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