3n^2+5n-10=0

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Solution for 3n^2+5n-10=0 equation: 


Simplifying
3n2 + 5n + -10 = 0

Reorder the terms:
-10 + 5n + 3n2 = 0

Solving
-10 + 5n + 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'.
-3.333333333 + 1.666666667n + n2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + 1.666666667n + 3.333333333 + n2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + 1.666666667n + n2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + 1.666666667n + n2 = 0 + 3.333333333
1.666666667n + n2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
1.666666667n + n2 = 3.333333333

The n term is 1.666666667n.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667n + 0.6944444447 + n2 = 3.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667n + n2 = 3.333333333 + 0.6944444447

Combine like terms: 3.333333333 + 0.6944444447 = 4.0277777777
0.6944444447 + 1.666666667n + n2 = 4.0277777777

Factor a perfect square on the left side:
(n + 0.8333333335)(n + 0.8333333335) = 4.0277777777

Calculate the square root of the right side: 2.00693243

Break this problem into two subproblems by setting 
(n + 0.8333333335) equal to 2.00693243 and -2.00693243.


Subproblem 1
n + 0.8333333335 = 2.00693243

Simplifying
n + 0.8333333335 = 2.00693243

Reorder the terms:
0.8333333335 + n = 2.00693243

Solving
0.8333333335 + n = 2.00693243

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + n = 2.00693243 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + n = 2.00693243 + -0.8333333335
n = 2.00693243 + -0.8333333335

Combine like terms: 2.00693243 + -0.8333333335 = 1.1735990965
n = 1.1735990965

Simplifying
n = 1.1735990965


Subproblem 2
n + 0.8333333335 = -2.00693243

Simplifying
n + 0.8333333335 = -2.00693243

Reorder the terms:
0.8333333335 + n = -2.00693243

Solving
0.8333333335 + n = -2.00693243

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + n = -2.00693243 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + n = -2.00693243 + -0.8333333335
n = -2.00693243 + -0.8333333335

Combine like terms: -2.00693243 + -0.8333333335 = -2.8402657635
n = -2.8402657635

Simplifying
n = -2.8402657635


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.1735990965, -2.8402657635}

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