3n^2=-9n+62

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Solution for 3n^2=-9n+62 equation:


Simplifying
3n2 = -9n + 62

Reorder the terms:
3n2 = 62 + -9n

Solving
3n2 = 62 + -9n

Solving for variable 'n'.

Reorder the terms:
-62 + 9n + 3n2 = 62 + -9n + -62 + 9n

Reorder the terms:
-62 + 9n + 3n2 = 62 + -62 + -9n + 9n

Combine like terms: 62 + -62 = 0
-62 + 9n + 3n2 = 0 + -9n + 9n
-62 + 9n + 3n2 = -9n + 9n

Combine like terms: -9n + 9n = 0
-62 + 9n + 3n2 = 0

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

Divide each side by '3'.
-20.66666667 + 3n + n2 = 0

Move the constant term to the right:

Add '20.66666667' to each side of the equation.
-20.66666667 + 3n + 20.66666667 + n2 = 0 + 20.66666667

Reorder the terms:
-20.66666667 + 20.66666667 + 3n + n2 = 0 + 20.66666667

Combine like terms: -20.66666667 + 20.66666667 = 0.00000000
0.00000000 + 3n + n2 = 0 + 20.66666667
3n + n2 = 0 + 20.66666667

Combine like terms: 0 + 20.66666667 = 20.66666667
3n + n2 = 20.66666667

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 = 20.66666667 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 20.66666667 + 2.25

Combine like terms: 20.66666667 + 2.25 = 22.91666667
2.25 + 3n + n2 = 22.91666667

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 22.91666667

Calculate the square root of the right side: 4.787135539

Break this problem into two subproblems by setting 
(n + 1.5) equal to 4.787135539 and -4.787135539.

Subproblem 1

n + 1.5 = 4.787135539 Simplifying n + 1.5 = 4.787135539 Reorder the terms: 1.5 + n = 4.787135539 Solving 1.5 + n = 4.787135539 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 = 4.787135539 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.787135539 + -1.5 n = 4.787135539 + -1.5 Combine like terms: 4.787135539 + -1.5 = 3.287135539 n = 3.287135539 Simplifying n = 3.287135539

Subproblem 2

n + 1.5 = -4.787135539 Simplifying n + 1.5 = -4.787135539 Reorder the terms: 1.5 + n = -4.787135539 Solving 1.5 + n = -4.787135539 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 = -4.787135539 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.787135539 + -1.5 n = -4.787135539 + -1.5 Combine like terms: -4.787135539 + -1.5 = -6.287135539 n = -6.287135539 Simplifying n = -6.287135539

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.287135539, -6.287135539}

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