3n(n+7)=-11-2(n)

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Solution for 3n(n+7)=-11-2(n) equation:


Simplifying
3n(n + 7) = -11 + -2(n)

Reorder the terms:
3n(7 + n) = -11 + -2(n)
(7 * 3n + n * 3n) = -11 + -2(n)
(21n + 3n2) = -11 + -2(n)

Solving
21n + 3n2 = -11 + -2n

Solving for variable 'n'.

Reorder the terms:
11 + 21n + 2n + 3n2 = -11 + -2n + 11 + 2n

Combine like terms: 21n + 2n = 23n
11 + 23n + 3n2 = -11 + -2n + 11 + 2n

Reorder the terms:
11 + 23n + 3n2 = -11 + 11 + -2n + 2n

Combine like terms: -11 + 11 = 0
11 + 23n + 3n2 = 0 + -2n + 2n
11 + 23n + 3n2 = -2n + 2n

Combine like terms: -2n + 2n = 0
11 + 23n + 3n2 = 0

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

Divide each side by '3'.
3.666666667 + 7.666666667n + n2 = 0

Move the constant term to the right:

Add '-3.666666667' to each side of the equation.
3.666666667 + 7.666666667n + -3.666666667 + n2 = 0 + -3.666666667

Reorder the terms:
3.666666667 + -3.666666667 + 7.666666667n + n2 = 0 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + 7.666666667n + n2 = 0 + -3.666666667
7.666666667n + n2 = 0 + -3.666666667

Combine like terms: 0 + -3.666666667 = -3.666666667
7.666666667n + n2 = -3.666666667

The n term is 7.666666667n.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667n + 14.69444445 + n2 = -3.666666667 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667n + n2 = -3.666666667 + 14.69444445

Combine like terms: -3.666666667 + 14.69444445 = 11.027777783
14.69444445 + 7.666666667n + n2 = 11.027777783

Factor a perfect square on the left side:
(n + 3.833333334)(n + 3.833333334) = 11.027777783

Calculate the square root of the right side: 3.320809808

Break this problem into two subproblems by setting 
(n + 3.833333334) equal to 3.320809808 and -3.320809808.

Subproblem 1

n + 3.833333334 = 3.320809808 Simplifying n + 3.833333334 = 3.320809808 Reorder the terms: 3.833333334 + n = 3.320809808 Solving 3.833333334 + n = 3.320809808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + n = 3.320809808 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + n = 3.320809808 + -3.833333334 n = 3.320809808 + -3.833333334 Combine like terms: 3.320809808 + -3.833333334 = -0.512523526 n = -0.512523526 Simplifying n = -0.512523526

Subproblem 2

n + 3.833333334 = -3.320809808 Simplifying n + 3.833333334 = -3.320809808 Reorder the terms: 3.833333334 + n = -3.320809808 Solving 3.833333334 + n = -3.320809808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + n = -3.320809808 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + n = -3.320809808 + -3.833333334 n = -3.320809808 + -3.833333334 Combine like terms: -3.320809808 + -3.833333334 = -7.154143142 n = -7.154143142 Simplifying n = -7.154143142

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.512523526, -7.154143142}

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