n^2+14n-3=4n-8

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Solution for n^2+14n-3=4n-8 equation:


Simplifying
n2 + 14n + -3 = 4n + -8

Reorder the terms:
-3 + 14n + n2 = 4n + -8

Reorder the terms:
-3 + 14n + n2 = -8 + 4n

Solving
-3 + 14n + n2 = -8 + 4n

Solving for variable 'n'.

Reorder the terms:
-3 + 8 + 14n + -4n + n2 = -8 + 4n + 8 + -4n

Combine like terms: -3 + 8 = 5
5 + 14n + -4n + n2 = -8 + 4n + 8 + -4n

Combine like terms: 14n + -4n = 10n
5 + 10n + n2 = -8 + 4n + 8 + -4n

Reorder the terms:
5 + 10n + n2 = -8 + 8 + 4n + -4n

Combine like terms: -8 + 8 = 0
5 + 10n + n2 = 0 + 4n + -4n
5 + 10n + n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
5 + 10n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 10n + -5 + n2 = 0 + -5

Reorder the terms:
5 + -5 + 10n + n2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 10n + n2 = 0 + -5
10n + n2 = 0 + -5

Combine like terms: 0 + -5 = -5
10n + n2 = -5

The n term is 10n.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10n + 25 + n2 = -5 + 25

Reorder the terms:
25 + 10n + n2 = -5 + 25

Combine like terms: -5 + 25 = 20
25 + 10n + n2 = 20

Factor a perfect square on the left side:
(n + 5)(n + 5) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(n + 5) equal to 4.472135955 and -4.472135955.

Subproblem 1

n + 5 = 4.472135955 Simplifying n + 5 = 4.472135955 Reorder the terms: 5 + n = 4.472135955 Solving 5 + n = 4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 4.472135955 + -5 Combine like terms: 5 + -5 = 0 0 + n = 4.472135955 + -5 n = 4.472135955 + -5 Combine like terms: 4.472135955 + -5 = -0.527864045 n = -0.527864045 Simplifying n = -0.527864045

Subproblem 2

n + 5 = -4.472135955 Simplifying n + 5 = -4.472135955 Reorder the terms: 5 + n = -4.472135955 Solving 5 + n = -4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -4.472135955 + -5 Combine like terms: 5 + -5 = 0 0 + n = -4.472135955 + -5 n = -4.472135955 + -5 Combine like terms: -4.472135955 + -5 = -9.472135955 n = -9.472135955 Simplifying n = -9.472135955

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.527864045, -9.472135955}

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