3n^2-10n-15=0

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Solution for 3n^2-10n-15=0 equation:


Simplifying
3n2 + -10n + -15 = 0

Reorder the terms:
-15 + -10n + 3n2 = 0

Solving
-15 + -10n + 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'.
-5 + -3.333333333n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -3.333333333n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -3.333333333n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -3.333333333n + n2 = 0 + 5
-3.333333333n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-3.333333333n + n2 = 5

The n term is -3.333333333n.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333n + 2.777777779 + n2 = 5 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333n + n2 = 5 + 2.777777779

Combine like terms: 5 + 2.777777779 = 7.777777779
2.777777779 + -3.333333333n + n2 = 7.777777779

Factor a perfect square on the left side:
(n + -1.666666667)(n + -1.666666667) = 7.777777779

Calculate the square root of the right side: 2.788866755

Break this problem into two subproblems by setting 
(n + -1.666666667) equal to 2.788866755 and -2.788866755.

Subproblem 1

n + -1.666666667 = 2.788866755 Simplifying n + -1.666666667 = 2.788866755 Reorder the terms: -1.666666667 + n = 2.788866755 Solving -1.666666667 + n = 2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + n = 2.788866755 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + n = 2.788866755 + 1.666666667 n = 2.788866755 + 1.666666667 Combine like terms: 2.788866755 + 1.666666667 = 4.455533422 n = 4.455533422 Simplifying n = 4.455533422

Subproblem 2

n + -1.666666667 = -2.788866755 Simplifying n + -1.666666667 = -2.788866755 Reorder the terms: -1.666666667 + n = -2.788866755 Solving -1.666666667 + n = -2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + n = -2.788866755 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + n = -2.788866755 + 1.666666667 n = -2.788866755 + 1.666666667 Combine like terms: -2.788866755 + 1.666666667 = -1.122200088 n = -1.122200088 Simplifying n = -1.122200088

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.455533422, -1.122200088}

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