6n+12=4n^2-10

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Solution for 6n+12=4n^2-10 equation: 


Simplifying
6n + 12 = 4n2 + -10

Reorder the terms:
12 + 6n = 4n2 + -10

Reorder the terms:
12 + 6n = -10 + 4n2

Solving
12 + 6n = -10 + 4n2

Solving for variable 'n'.

Reorder the terms:
12 + 10 + 6n + -4n2 = -10 + 4n2 + 10 + -4n2

Combine like terms: 12 + 10 = 22
22 + 6n + -4n2 = -10 + 4n2 + 10 + -4n2

Reorder the terms:
22 + 6n + -4n2 = -10 + 10 + 4n2 + -4n2

Combine like terms: -10 + 10 = 0
22 + 6n + -4n2 = 0 + 4n2 + -4n2
22 + 6n + -4n2 = 4n2 + -4n2

Combine like terms: 4n2 + -4n2 = 0
22 + 6n + -4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(11 + 3n + -2n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(11 + 3n + -2n2)' equal to zero and attempt to solve:

Simplifying
11 + 3n + -2n2 = 0

Solving
11 + 3n + -2n2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-5.5 + -1.5n + n2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + -1.5n + 5.5 + n2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + -1.5n + n2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + -1.5n + n2 = 0 + 5.5
-1.5n + n2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
-1.5n + n2 = 5.5

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = 5.5 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 5.5 + 0.5625

Combine like terms: 5.5 + 0.5625 = 6.0625
0.5625 + -1.5n + n2 = 6.0625

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(n + -0.75) equal to 2.46221445 and -2.46221445.


Subproblem 1
n + -0.75 = 2.46221445

Simplifying
n + -0.75 = 2.46221445

Reorder the terms:
-0.75 + n = 2.46221445

Solving
-0.75 + n = 2.46221445

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '0.75' to each side of the equation.
-0.75 + 0.75 + n = 2.46221445 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + n = 2.46221445 + 0.75
n = 2.46221445 + 0.75

Combine like terms: 2.46221445 + 0.75 = 3.21221445
n = 3.21221445

Simplifying
n = 3.21221445


Subproblem 2
n + -0.75 = -2.46221445

Simplifying
n + -0.75 = -2.46221445

Reorder the terms:
-0.75 + n = -2.46221445

Solving
-0.75 + n = -2.46221445

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '0.75' to each side of the equation.
-0.75 + 0.75 + n = -2.46221445 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + n = -2.46221445 + 0.75
n = -2.46221445 + 0.75

Combine like terms: -2.46221445 + 0.75 = -1.71221445
n = -1.71221445

Simplifying
n = -1.71221445


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.21221445, -1.71221445}
Solution
n = {3.21221445, -1.71221445}

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