2n(5n+3)+7(5n-3)=

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Solution for 2n(5n+3)+7(5n-3)= equation: 


Simplifying
2n(5n + 3) + 7(5n + -3) = 0

Reorder the terms:
2n(3 + 5n) + 7(5n + -3) = 0
(3 * 2n + 5n * 2n) + 7(5n + -3) = 0
(6n + 10n2) + 7(5n + -3) = 0

Reorder the terms:
6n + 10n2 + 7(-3 + 5n) = 0
6n + 10n2 + (-3 * 7 + 5n * 7) = 0
6n + 10n2 + (-21 + 35n) = 0

Reorder the terms:
-21 + 6n + 35n + 10n2 = 0

Combine like terms: 6n + 35n = 41n
-21 + 41n + 10n2 = 0

Solving
-21 + 41n + 10n2 = 0

Solving for variable 'n'.

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

Divide each side by '10'.
-2.1 + 4.1n + n2 = 0.0

Move the constant term to the right:

Add '2.1' to each side of the equation.
-2.1 + 4.1n + 2.1 + n2 = 0.0 + 2.1

Reorder the terms:
-2.1 + 2.1 + 4.1n + n2 = 0.0 + 2.1

Combine like terms: -2.1 + 2.1 = 0.0
0.0 + 4.1n + n2 = 0.0 + 2.1
4.1n + n2 = 0.0 + 2.1

Combine like terms: 0.0 + 2.1 = 2.1
4.1n + n2 = 2.1

The n term is 4.1n.  Take half its coefficient (2.05).
Square it (4.2025) and add it to both sides.

Add '4.2025' to each side of the equation.
4.1n + 4.2025 + n2 = 2.1 + 4.2025

Reorder the terms:
4.2025 + 4.1n + n2 = 2.1 + 4.2025

Combine like terms: 2.1 + 4.2025 = 6.3025
4.2025 + 4.1n + n2 = 6.3025

Factor a perfect square on the left side:
(n + 2.05)(n + 2.05) = 6.3025

Calculate the square root of the right side: 2.510478042

Break this problem into two subproblems by setting 
(n + 2.05) equal to 2.510478042 and -2.510478042.


Subproblem 1
n + 2.05 = 2.510478042

Simplifying
n + 2.05 = 2.510478042

Reorder the terms:
2.05 + n = 2.510478042

Solving
2.05 + n = 2.510478042

Solving for variable 'n'.

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

Add '-2.05' to each side of the equation.
2.05 + -2.05 + n = 2.510478042 + -2.05

Combine like terms: 2.05 + -2.05 = 0.00
0.00 + n = 2.510478042 + -2.05
n = 2.510478042 + -2.05

Combine like terms: 2.510478042 + -2.05 = 0.460478042
n = 0.460478042

Simplifying
n = 0.460478042


Subproblem 2
n + 2.05 = -2.510478042

Simplifying
n + 2.05 = -2.510478042

Reorder the terms:
2.05 + n = -2.510478042

Solving
2.05 + n = -2.510478042

Solving for variable 'n'.

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

Add '-2.05' to each side of the equation.
2.05 + -2.05 + n = -2.510478042 + -2.05

Combine like terms: 2.05 + -2.05 = 0.00
0.00 + n = -2.510478042 + -2.05
n = -2.510478042 + -2.05

Combine like terms: -2.510478042 + -2.05 = -4.560478042
n = -4.560478042

Simplifying
n = -4.560478042


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.460478042, -4.560478042}

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