5n^2-22=-3n

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Solution for 5n^2-22=-3n equation:


Simplifying
5n2 + -22 = -3n

Reorder the terms:
-22 + 5n2 = -3n

Solving
-22 + 5n2 = -3n

Solving for variable 'n'.

Reorder the terms:
-22 + 3n + 5n2 = -3n + 3n

Combine like terms: -3n + 3n = 0
-22 + 3n + 5n2 = 0

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

Divide each side by '5'.
-4.4 + 0.6n + n2 = 0

Move the constant term to the right:

Add '4.4' to each side of the equation.
-4.4 + 0.6n + 4.4 + n2 = 0 + 4.4

Reorder the terms:
-4.4 + 4.4 + 0.6n + n2 = 0 + 4.4

Combine like terms: -4.4 + 4.4 = 0.0
0.0 + 0.6n + n2 = 0 + 4.4
0.6n + n2 = 0 + 4.4

Combine like terms: 0 + 4.4 = 4.4
0.6n + n2 = 4.4

The n term is 0.6n.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6n + 0.09 + n2 = 4.4 + 0.09

Reorder the terms:
0.09 + 0.6n + n2 = 4.4 + 0.09

Combine like terms: 4.4 + 0.09 = 4.49
0.09 + 0.6n + n2 = 4.49

Factor a perfect square on the left side:
(n + 0.3)(n + 0.3) = 4.49

Calculate the square root of the right side: 2.11896201

Break this problem into two subproblems by setting 
(n + 0.3) equal to 2.11896201 and -2.11896201.

Subproblem 1

n + 0.3 = 2.11896201 Simplifying n + 0.3 = 2.11896201 Reorder the terms: 0.3 + n = 2.11896201 Solving 0.3 + n = 2.11896201 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = 2.11896201 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = 2.11896201 + -0.3 n = 2.11896201 + -0.3 Combine like terms: 2.11896201 + -0.3 = 1.81896201 n = 1.81896201 Simplifying n = 1.81896201

Subproblem 2

n + 0.3 = -2.11896201 Simplifying n + 0.3 = -2.11896201 Reorder the terms: 0.3 + n = -2.11896201 Solving 0.3 + n = -2.11896201 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = -2.11896201 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = -2.11896201 + -0.3 n = -2.11896201 + -0.3 Combine like terms: -2.11896201 + -0.3 = -2.41896201 n = -2.41896201 Simplifying n = -2.41896201

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.81896201, -2.41896201}

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