2n^2-6n-3=(4n-5)

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Solution for 2n^2-6n-3=(4n-5) equation: 


Simplifying
2n2 + -6n + -3 = (4n + -5)

Reorder the terms:
-3 + -6n + 2n2 = (4n + -5)

Reorder the terms:
-3 + -6n + 2n2 = (-5 + 4n)

Remove parenthesis around (-5 + 4n)
-3 + -6n + 2n2 = -5 + 4n

Solving
-3 + -6n + 2n2 = -5 + 4n

Solving for variable 'n'.

Reorder the terms:
-3 + 5 + -6n + -4n + 2n2 = -5 + 4n + 5 + -4n

Combine like terms: -3 + 5 = 2
2 + -6n + -4n + 2n2 = -5 + 4n + 5 + -4n

Combine like terms: -6n + -4n = -10n
2 + -10n + 2n2 = -5 + 4n + 5 + -4n

Reorder the terms:
2 + -10n + 2n2 = -5 + 5 + 4n + -4n

Combine like terms: -5 + 5 = 0
2 + -10n + 2n2 = 0 + 4n + -4n
2 + -10n + 2n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
2 + -10n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + -5n + n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(1 + -5n + n2)' equal to zero and attempt to solve:

Simplifying
1 + -5n + n2 = 0

Solving
1 + -5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -5n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5n + n2 = 0 + -1
-5n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5n + n2 = -1

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = -1 + 6.25

Reorder the terms:
6.25 + -5n + n2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5n + n2 = 5.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(n + -2.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
n + -2.5 = 2.291287847

Simplifying
n + -2.5 = 2.291287847

Reorder the terms:
-2.5 + n = 2.291287847

Solving
-2.5 + n = 2.291287847

Solving for variable 'n'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = 2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = 2.291287847 + 2.5
n = 2.291287847 + 2.5

Combine like terms: 2.291287847 + 2.5 = 4.791287847
n = 4.791287847

Simplifying
n = 4.791287847


Subproblem 2
n + -2.5 = -2.291287847

Simplifying
n + -2.5 = -2.291287847

Reorder the terms:
-2.5 + n = -2.291287847

Solving
-2.5 + n = -2.291287847

Solving for variable 'n'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = -2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = -2.291287847 + 2.5
n = -2.291287847 + 2.5

Combine like terms: -2.291287847 + 2.5 = 0.208712153
n = 0.208712153

Simplifying
n = 0.208712153


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.791287847, 0.208712153}
Solution
n = {4.791287847, 0.208712153}

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