13n^2=12n^2-40n-200

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Solution for 13n^2=12n^2-40n-200 equation:


Simplifying
13n2 = 12n2 + -40n + -200

Reorder the terms:
13n2 = -200 + -40n + 12n2

Solving
13n2 = -200 + -40n + 12n2

Solving for variable 'n'.

Reorder the terms:
200 + 40n + 13n2 + -12n2 = -200 + -40n + 12n2 + 200 + 40n + -12n2

Combine like terms: 13n2 + -12n2 = 1n2
200 + 40n + 1n2 = -200 + -40n + 12n2 + 200 + 40n + -12n2

Reorder the terms:
200 + 40n + 1n2 = -200 + 200 + -40n + 40n + 12n2 + -12n2

Combine like terms: -200 + 200 = 0
200 + 40n + 1n2 = 0 + -40n + 40n + 12n2 + -12n2
200 + 40n + 1n2 = -40n + 40n + 12n2 + -12n2

Combine like terms: -40n + 40n = 0
200 + 40n + 1n2 = 0 + 12n2 + -12n2
200 + 40n + 1n2 = 12n2 + -12n2

Combine like terms: 12n2 + -12n2 = 0
200 + 40n + 1n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-200' to each side of the equation.
200 + 40n + -200 + n2 = 0 + -200

Reorder the terms:
200 + -200 + 40n + n2 = 0 + -200

Combine like terms: 200 + -200 = 0
0 + 40n + n2 = 0 + -200
40n + n2 = 0 + -200

Combine like terms: 0 + -200 = -200
40n + n2 = -200

The n term is 40n.  Take half its coefficient (20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
40n + 400 + n2 = -200 + 400

Reorder the terms:
400 + 40n + n2 = -200 + 400

Combine like terms: -200 + 400 = 200
400 + 40n + n2 = 200

Factor a perfect square on the left side:
(n + 20)(n + 20) = 200

Calculate the square root of the right side: 14.142135624

Break this problem into two subproblems by setting 
(n + 20) equal to 14.142135624 and -14.142135624.

Subproblem 1

n + 20 = 14.142135624 Simplifying n + 20 = 14.142135624 Reorder the terms: 20 + n = 14.142135624 Solving 20 + n = 14.142135624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-20' to each side of the equation. 20 + -20 + n = 14.142135624 + -20 Combine like terms: 20 + -20 = 0 0 + n = 14.142135624 + -20 n = 14.142135624 + -20 Combine like terms: 14.142135624 + -20 = -5.857864376 n = -5.857864376 Simplifying n = -5.857864376

Subproblem 2

n + 20 = -14.142135624 Simplifying n + 20 = -14.142135624 Reorder the terms: 20 + n = -14.142135624 Solving 20 + n = -14.142135624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-20' to each side of the equation. 20 + -20 + n = -14.142135624 + -20 Combine like terms: 20 + -20 = 0 0 + n = -14.142135624 + -20 n = -14.142135624 + -20 Combine like terms: -14.142135624 + -20 = -34.142135624 n = -34.142135624 Simplifying n = -34.142135624

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-5.857864376, -34.142135624}

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