n^2+3n-22=-2

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


Simplifying
n2 + 3n + -22 = -2

Reorder the terms:
-22 + 3n + n2 = -2

Solving
-22 + 3n + n2 = -2

Solving for variable 'n'.

Reorder the terms:
-22 + 2 + 3n + n2 = -2 + 2

Combine like terms: -22 + 2 = -20
-20 + 3n + n2 = -2 + 2

Combine like terms: -2 + 2 = 0
-20 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 3n + 20 + n2 = 0 + 20

Reorder the terms:
-20 + 20 + 3n + n2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 3n + n2 = 0 + 20
3n + n2 = 0 + 20

Combine like terms: 0 + 20 = 20
3n + n2 = 20

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 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 20 + 2.25

Combine like terms: 20 + 2.25 = 22.25
2.25 + 3n + n2 = 22.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(n + 1.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

n + 1.5 = 4.716990566 Simplifying n + 1.5 = 4.716990566 Reorder the terms: 1.5 + n = 4.716990566 Solving 1.5 + n = 4.716990566 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 = 4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.716990566 + -1.5 n = 4.716990566 + -1.5 Combine like terms: 4.716990566 + -1.5 = 3.216990566 n = 3.216990566 Simplifying n = 3.216990566

Subproblem 2

n + 1.5 = -4.716990566 Simplifying n + 1.5 = -4.716990566 Reorder the terms: 1.5 + n = -4.716990566 Solving 1.5 + n = -4.716990566 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 = -4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.716990566 + -1.5 n = -4.716990566 + -1.5 Combine like terms: -4.716990566 + -1.5 = -6.216990566 n = -6.216990566 Simplifying n = -6.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.216990566, -6.216990566}

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