3n^2-12n=19

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Solution for 3n^2-12n=19 equation:


Simplifying
3n2 + -12n = 19

Reorder the terms:
-12n + 3n2 = 19

Solving
-12n + 3n2 = 19

Solving for variable 'n'.

Reorder the terms:
-19 + -12n + 3n2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + -12n + 3n2 = 0

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

Divide each side by '3'.
-6.333333333 + -4n + n2 = 0

Move the constant term to the right:

Add '6.333333333' to each side of the equation.
-6.333333333 + -4n + 6.333333333 + n2 = 0 + 6.333333333

Reorder the terms:
-6.333333333 + 6.333333333 + -4n + n2 = 0 + 6.333333333

Combine like terms: -6.333333333 + 6.333333333 = 0.000000000
0.000000000 + -4n + n2 = 0 + 6.333333333
-4n + n2 = 0 + 6.333333333

Combine like terms: 0 + 6.333333333 = 6.333333333
-4n + n2 = 6.333333333

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 6.333333333 + 4

Reorder the terms:
4 + -4n + n2 = 6.333333333 + 4

Combine like terms: 6.333333333 + 4 = 10.333333333
4 + -4n + n2 = 10.333333333

Factor a perfect square on the left side:
(n + -2)(n + -2) = 10.333333333

Calculate the square root of the right side: 3.214550254

Break this problem into two subproblems by setting 
(n + -2) equal to 3.214550254 and -3.214550254.

Subproblem 1

n + -2 = 3.214550254 Simplifying n + -2 = 3.214550254 Reorder the terms: -2 + n = 3.214550254 Solving -2 + n = 3.214550254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 3.214550254 + 2 Combine like terms: -2 + 2 = 0 0 + n = 3.214550254 + 2 n = 3.214550254 + 2 Combine like terms: 3.214550254 + 2 = 5.214550254 n = 5.214550254 Simplifying n = 5.214550254

Subproblem 2

n + -2 = -3.214550254 Simplifying n + -2 = -3.214550254 Reorder the terms: -2 + n = -3.214550254 Solving -2 + n = -3.214550254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -3.214550254 + 2 Combine like terms: -2 + 2 = 0 0 + n = -3.214550254 + 2 n = -3.214550254 + 2 Combine like terms: -3.214550254 + 2 = -1.214550254 n = -1.214550254 Simplifying n = -1.214550254

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.214550254, -1.214550254}

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