n+(3n-15)(n+20)=

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Solution for n+(3n-15)(n+20)= equation:


Simplifying
n + (3n + -15)(n + 20) = 0

Reorder the terms:
n + (-15 + 3n)(n + 20) = 0

Reorder the terms:
n + (-15 + 3n)(20 + n) = 0

Multiply (-15 + 3n) * (20 + n)
n + (-15(20 + n) + 3n * (20 + n)) = 0
n + ((20 * -15 + n * -15) + 3n * (20 + n)) = 0
n + ((-300 + -15n) + 3n * (20 + n)) = 0
n + (-300 + -15n + (20 * 3n + n * 3n)) = 0
n + (-300 + -15n + (60n + 3n2)) = 0

Combine like terms: -15n + 60n = 45n
n + (-300 + 45n + 3n2) = 0

Reorder the terms:
-300 + n + 45n + 3n2 = 0

Combine like terms: n + 45n = 46n
-300 + 46n + 3n2 = 0

Solving
-300 + 46n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-100 + 15.33333333n + n2 = 0

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + 15.33333333n + 100 + n2 = 0 + 100

Reorder the terms:
-100 + 100 + 15.33333333n + n2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + 15.33333333n + n2 = 0 + 100
15.33333333n + n2 = 0 + 100

Combine like terms: 0 + 100 = 100
15.33333333n + n2 = 100

The n term is 15.33333333n.  Take half its coefficient (7.666666665).
Square it (58.77777775) and add it to both sides.

Add '58.77777775' to each side of the equation.
15.33333333n + 58.77777775 + n2 = 100 + 58.77777775

Reorder the terms:
58.77777775 + 15.33333333n + n2 = 100 + 58.77777775

Combine like terms: 100 + 58.77777775 = 158.77777775
58.77777775 + 15.33333333n + n2 = 158.77777775

Factor a perfect square on the left side:
(n + 7.666666665)(n + 7.666666665) = 158.77777775

Calculate the square root of the right side: 12.600705447

Break this problem into two subproblems by setting 
(n + 7.666666665) equal to 12.600705447 and -12.600705447.

Subproblem 1

n + 7.666666665 = 12.600705447 Simplifying n + 7.666666665 = 12.600705447 Reorder the terms: 7.666666665 + n = 12.600705447 Solving 7.666666665 + n = 12.600705447 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.666666665' to each side of the equation. 7.666666665 + -7.666666665 + n = 12.600705447 + -7.666666665 Combine like terms: 7.666666665 + -7.666666665 = 0.000000000 0.000000000 + n = 12.600705447 + -7.666666665 n = 12.600705447 + -7.666666665 Combine like terms: 12.600705447 + -7.666666665 = 4.934038782 n = 4.934038782 Simplifying n = 4.934038782

Subproblem 2

n + 7.666666665 = -12.600705447 Simplifying n + 7.666666665 = -12.600705447 Reorder the terms: 7.666666665 + n = -12.600705447 Solving 7.666666665 + n = -12.600705447 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.666666665' to each side of the equation. 7.666666665 + -7.666666665 + n = -12.600705447 + -7.666666665 Combine like terms: 7.666666665 + -7.666666665 = 0.000000000 0.000000000 + n = -12.600705447 + -7.666666665 n = -12.600705447 + -7.666666665 Combine like terms: -12.600705447 + -7.666666665 = -20.267372112 n = -20.267372112 Simplifying n = -20.267372112

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.934038782, -20.267372112}

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