5n^2+11n=14

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Solution for 5n^2+11n=14 equation:


Simplifying
5n2 + 11n = 14

Reorder the terms:
11n + 5n2 = 14

Solving
11n + 5n2 = 14

Solving for variable 'n'.

Reorder the terms:
-14 + 11n + 5n2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + 11n + 5n2 = 0

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

Divide each side by '5'.
-2.8 + 2.2n + n2 = 0

Move the constant term to the right:

Add '2.8' to each side of the equation.
-2.8 + 2.2n + 2.8 + n2 = 0 + 2.8

Reorder the terms:
-2.8 + 2.8 + 2.2n + n2 = 0 + 2.8

Combine like terms: -2.8 + 2.8 = 0.0
0.0 + 2.2n + n2 = 0 + 2.8
2.2n + n2 = 0 + 2.8

Combine like terms: 0 + 2.8 = 2.8
2.2n + n2 = 2.8

The n term is 2.2n.  Take half its coefficient (1.1).
Square it (1.21) and add it to both sides.

Add '1.21' to each side of the equation.
2.2n + 1.21 + n2 = 2.8 + 1.21

Reorder the terms:
1.21 + 2.2n + n2 = 2.8 + 1.21

Combine like terms: 2.8 + 1.21 = 4.01
1.21 + 2.2n + n2 = 4.01

Factor a perfect square on the left side:
(n + 1.1)(n + 1.1) = 4.01

Calculate the square root of the right side: 2.002498439

Break this problem into two subproblems by setting 
(n + 1.1) equal to 2.002498439 and -2.002498439.

Subproblem 1

n + 1.1 = 2.002498439 Simplifying n + 1.1 = 2.002498439 Reorder the terms: 1.1 + n = 2.002498439 Solving 1.1 + n = 2.002498439 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.1' to each side of the equation. 1.1 + -1.1 + n = 2.002498439 + -1.1 Combine like terms: 1.1 + -1.1 = 0.0 0.0 + n = 2.002498439 + -1.1 n = 2.002498439 + -1.1 Combine like terms: 2.002498439 + -1.1 = 0.902498439 n = 0.902498439 Simplifying n = 0.902498439

Subproblem 2

n + 1.1 = -2.002498439 Simplifying n + 1.1 = -2.002498439 Reorder the terms: 1.1 + n = -2.002498439 Solving 1.1 + n = -2.002498439 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.1' to each side of the equation. 1.1 + -1.1 + n = -2.002498439 + -1.1 Combine like terms: 1.1 + -1.1 = 0.0 0.0 + n = -2.002498439 + -1.1 n = -2.002498439 + -1.1 Combine like terms: -2.002498439 + -1.1 = -3.102498439 n = -3.102498439 Simplifying n = -3.102498439

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.902498439, -3.102498439}

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