5n^2+6n=5

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


Simplifying
5n2 + 6n = 5

Reorder the terms:
6n + 5n2 = 5

Solving
6n + 5n2 = 5

Solving for variable 'n'.

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

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

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

Divide each side by '5'.
-1 + 1.2n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.2n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.2n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.2n + n2 = 0 + 1
1.2n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.2n + n2 = 1

The n term is 1.2n.  Take half its coefficient (0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
1.2n + 0.36 + n2 = 1 + 0.36

Reorder the terms:
0.36 + 1.2n + n2 = 1 + 0.36

Combine like terms: 1 + 0.36 = 1.36
0.36 + 1.2n + n2 = 1.36

Factor a perfect square on the left side:
(n + 0.6)(n + 0.6) = 1.36

Calculate the square root of the right side: 1.166190379

Break this problem into two subproblems by setting 
(n + 0.6) equal to 1.166190379 and -1.166190379.

Subproblem 1

n + 0.6 = 1.166190379 Simplifying n + 0.6 = 1.166190379 Reorder the terms: 0.6 + n = 1.166190379 Solving 0.6 + n = 1.166190379 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + n = 1.166190379 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + n = 1.166190379 + -0.6 n = 1.166190379 + -0.6 Combine like terms: 1.166190379 + -0.6 = 0.566190379 n = 0.566190379 Simplifying n = 0.566190379

Subproblem 2

n + 0.6 = -1.166190379 Simplifying n + 0.6 = -1.166190379 Reorder the terms: 0.6 + n = -1.166190379 Solving 0.6 + n = -1.166190379 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + n = -1.166190379 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + n = -1.166190379 + -0.6 n = -1.166190379 + -0.6 Combine like terms: -1.166190379 + -0.6 = -1.766190379 n = -1.766190379 Simplifying n = -1.766190379

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.566190379, -1.766190379}

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