n(n+3)=85

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Solution for n(n+3)=85 equation:


Simplifying
n(n + 3) = 85

Reorder the terms:
n(3 + n) = 85
(3 * n + n * n) = 85
(3n + n2) = 85

Solving
3n + n2 = 85

Solving for variable 'n'.

Reorder the terms:
-85 + 3n + n2 = 85 + -85

Combine like terms: 85 + -85 = 0
-85 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '85' to each side of the equation.
-85 + 3n + 85 + n2 = 0 + 85

Reorder the terms:
-85 + 85 + 3n + n2 = 0 + 85

Combine like terms: -85 + 85 = 0
0 + 3n + n2 = 0 + 85
3n + n2 = 0 + 85

Combine like terms: 0 + 85 = 85
3n + n2 = 85

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 85 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 85 + 2.25

Combine like terms: 85 + 2.25 = 87.25
2.25 + 3n + n2 = 87.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 87.25

Calculate the square root of the right side: 9.340770846

Break this problem into two subproblems by setting 
(n + 1.5) equal to 9.340770846 and -9.340770846.

Subproblem 1

n + 1.5 = 9.340770846 Simplifying n + 1.5 = 9.340770846 Reorder the terms: 1.5 + n = 9.340770846 Solving 1.5 + n = 9.340770846 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 9.340770846 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 9.340770846 + -1.5 n = 9.340770846 + -1.5 Combine like terms: 9.340770846 + -1.5 = 7.840770846 n = 7.840770846 Simplifying n = 7.840770846

Subproblem 2

n + 1.5 = -9.340770846 Simplifying n + 1.5 = -9.340770846 Reorder the terms: 1.5 + n = -9.340770846 Solving 1.5 + n = -9.340770846 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -9.340770846 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -9.340770846 + -1.5 n = -9.340770846 + -1.5 Combine like terms: -9.340770846 + -1.5 = -10.840770846 n = -10.840770846 Simplifying n = -10.840770846

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.840770846, -10.840770846}

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