6n+19=4+n^2

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Solution for 6n+19=4+n^2 equation:


Simplifying
6n + 19 = 4 + n2

Reorder the terms:
19 + 6n = 4 + n2

Solving
19 + 6n = 4 + n2

Solving for variable 'n'.

Reorder the terms:
19 + -4 + 6n + -1n2 = 4 + n2 + -4 + -1n2

Combine like terms: 19 + -4 = 15
15 + 6n + -1n2 = 4 + n2 + -4 + -1n2

Reorder the terms:
15 + 6n + -1n2 = 4 + -4 + n2 + -1n2

Combine like terms: 4 + -4 = 0
15 + 6n + -1n2 = 0 + n2 + -1n2
15 + 6n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
15 + 6n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-15 + -6n + n2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -6n + 15 + n2 = 0 + 15

Reorder the terms:
-15 + 15 + -6n + n2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -6n + n2 = 0 + 15
-6n + n2 = 0 + 15

Combine like terms: 0 + 15 = 15
-6n + n2 = 15

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 = 15 + 9

Reorder the terms:
9 + -6n + n2 = 15 + 9

Combine like terms: 15 + 9 = 24
9 + -6n + n2 = 24

Factor a perfect square on the left side:
(n + -3)(n + -3) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(n + -3) equal to 4.898979486 and -4.898979486.

Subproblem 1

n + -3 = 4.898979486 Simplifying n + -3 = 4.898979486 Reorder the terms: -3 + n = 4.898979486 Solving -3 + n = 4.898979486 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 = 4.898979486 + 3 Combine like terms: -3 + 3 = 0 0 + n = 4.898979486 + 3 n = 4.898979486 + 3 Combine like terms: 4.898979486 + 3 = 7.898979486 n = 7.898979486 Simplifying n = 7.898979486

Subproblem 2

n + -3 = -4.898979486 Simplifying n + -3 = -4.898979486 Reorder the terms: -3 + n = -4.898979486 Solving -3 + n = -4.898979486 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 = -4.898979486 + 3 Combine like terms: -3 + 3 = 0 0 + n = -4.898979486 + 3 n = -4.898979486 + 3 Combine like terms: -4.898979486 + 3 = -1.898979486 n = -1.898979486 Simplifying n = -1.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.898979486, -1.898979486}

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