6n^2-5n-12=18

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Solution for 6n^2-5n-12=18 equation:


Simplifying
6n2 + -5n + -12 = 18

Reorder the terms:
-12 + -5n + 6n2 = 18

Solving
-12 + -5n + 6n2 = 18

Solving for variable 'n'.

Reorder the terms:
-12 + -18 + -5n + 6n2 = 18 + -18

Combine like terms: -12 + -18 = -30
-30 + -5n + 6n2 = 18 + -18

Combine like terms: 18 + -18 = 0
-30 + -5n + 6n2 = 0

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

Divide each side by '6'.
-5 + -0.8333333333n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.8333333333n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.8333333333n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.8333333333n + n2 = 0 + 5
-0.8333333333n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.8333333333n + n2 = 5

The n term is -0.8333333333n.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333n + 0.1736111111 + n2 = 5 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333n + n2 = 5 + 0.1736111111

Combine like terms: 5 + 0.1736111111 = 5.1736111111
0.1736111111 + -0.8333333333n + n2 = 5.1736111111

Factor a perfect square on the left side:
(n + -0.4166666667)(n + -0.4166666667) = 5.1736111111

Calculate the square root of the right side: 2.274557344

Break this problem into two subproblems by setting 
(n + -0.4166666667) equal to 2.274557344 and -2.274557344.

Subproblem 1

n + -0.4166666667 = 2.274557344 Simplifying n + -0.4166666667 = 2.274557344 Reorder the terms: -0.4166666667 + n = 2.274557344 Solving -0.4166666667 + n = 2.274557344 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + n = 2.274557344 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + n = 2.274557344 + 0.4166666667 n = 2.274557344 + 0.4166666667 Combine like terms: 2.274557344 + 0.4166666667 = 2.6912240107 n = 2.6912240107 Simplifying n = 2.6912240107

Subproblem 2

n + -0.4166666667 = -2.274557344 Simplifying n + -0.4166666667 = -2.274557344 Reorder the terms: -0.4166666667 + n = -2.274557344 Solving -0.4166666667 + n = -2.274557344 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + n = -2.274557344 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + n = -2.274557344 + 0.4166666667 n = -2.274557344 + 0.4166666667 Combine like terms: -2.274557344 + 0.4166666667 = -1.8578906773 n = -1.8578906773 Simplifying n = -1.8578906773

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.6912240107, -1.8578906773}

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