3n^2-6n-47=-10

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Solution for 3n^2-6n-47=-10 equation:


Simplifying
3n2 + -6n + -47 = -10

Reorder the terms:
-47 + -6n + 3n2 = -10

Solving
-47 + -6n + 3n2 = -10

Solving for variable 'n'.

Reorder the terms:
-47 + 10 + -6n + 3n2 = -10 + 10

Combine like terms: -47 + 10 = -37
-37 + -6n + 3n2 = -10 + 10

Combine like terms: -10 + 10 = 0
-37 + -6n + 3n2 = 0

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

Divide each side by '3'.
-12.33333333 + -2n + n2 = 0

Move the constant term to the right:

Add '12.33333333' to each side of the equation.
-12.33333333 + -2n + 12.33333333 + n2 = 0 + 12.33333333

Reorder the terms:
-12.33333333 + 12.33333333 + -2n + n2 = 0 + 12.33333333

Combine like terms: -12.33333333 + 12.33333333 = 0.00000000
0.00000000 + -2n + n2 = 0 + 12.33333333
-2n + n2 = 0 + 12.33333333

Combine like terms: 0 + 12.33333333 = 12.33333333
-2n + n2 = 12.33333333

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 12.33333333 + 1

Reorder the terms:
1 + -2n + n2 = 12.33333333 + 1

Combine like terms: 12.33333333 + 1 = 13.33333333
1 + -2n + n2 = 13.33333333

Factor a perfect square on the left side:
(n + -1)(n + -1) = 13.33333333

Calculate the square root of the right side: 3.651483716

Break this problem into two subproblems by setting 
(n + -1) equal to 3.651483716 and -3.651483716.

Subproblem 1

n + -1 = 3.651483716 Simplifying n + -1 = 3.651483716 Reorder the terms: -1 + n = 3.651483716 Solving -1 + n = 3.651483716 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 3.651483716 + 1 Combine like terms: -1 + 1 = 0 0 + n = 3.651483716 + 1 n = 3.651483716 + 1 Combine like terms: 3.651483716 + 1 = 4.651483716 n = 4.651483716 Simplifying n = 4.651483716

Subproblem 2

n + -1 = -3.651483716 Simplifying n + -1 = -3.651483716 Reorder the terms: -1 + n = -3.651483716 Solving -1 + n = -3.651483716 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -3.651483716 + 1 Combine like terms: -1 + 1 = 0 0 + n = -3.651483716 + 1 n = -3.651483716 + 1 Combine like terms: -3.651483716 + 1 = -2.651483716 n = -2.651483716 Simplifying n = -2.651483716

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.651483716, -2.651483716}

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