n^2+3n-10=-5n^2-3

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


Simplifying
n2 + 3n + -10 = -5n2 + -3

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

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

Solving
-10 + 3n + n2 = -3 + -5n2

Solving for variable 'n'.

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

Combine like terms: -10 + 3 = -7
-7 + 3n + n2 + 5n2 = -3 + -5n2 + 3 + 5n2

Combine like terms: n2 + 5n2 = 6n2
-7 + 3n + 6n2 = -3 + -5n2 + 3 + 5n2

Reorder the terms:
-7 + 3n + 6n2 = -3 + 3 + -5n2 + 5n2

Combine like terms: -3 + 3 = 0
-7 + 3n + 6n2 = 0 + -5n2 + 5n2
-7 + 3n + 6n2 = -5n2 + 5n2

Combine like terms: -5n2 + 5n2 = 0
-7 + 3n + 6n2 = 0

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

Divide each side by '6'.
-1.166666667 + 0.5n + n2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + 0.5n + 1.166666667 + n2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + 0.5n + n2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + 0.5n + n2 = 0 + 1.166666667
0.5n + n2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
0.5n + n2 = 1.166666667

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 1.166666667 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 1.166666667 + 0.0625

Combine like terms: 1.166666667 + 0.0625 = 1.229166667
0.0625 + 0.5n + n2 = 1.229166667

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 1.229166667

Calculate the square root of the right side: 1.108677891

Break this problem into two subproblems by setting 
(n + 0.25) equal to 1.108677891 and -1.108677891.

Subproblem 1

n + 0.25 = 1.108677891 Simplifying n + 0.25 = 1.108677891 Reorder the terms: 0.25 + n = 1.108677891 Solving 0.25 + n = 1.108677891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.108677891 + -0.25 n = 1.108677891 + -0.25 Combine like terms: 1.108677891 + -0.25 = 0.858677891 n = 0.858677891 Simplifying n = 0.858677891

Subproblem 2

n + 0.25 = -1.108677891 Simplifying n + 0.25 = -1.108677891 Reorder the terms: 0.25 + n = -1.108677891 Solving 0.25 + n = -1.108677891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.108677891 + -0.25 n = -1.108677891 + -0.25 Combine like terms: -1.108677891 + -0.25 = -1.358677891 n = -1.358677891 Simplifying n = -1.358677891

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.858677891, -1.358677891}

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