n^2-3n-18=11

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Solution for n^2-3n-18=11 equation:


Simplifying
n2 + -3n + -18 = 11

Reorder the terms:
-18 + -3n + n2 = 11

Solving
-18 + -3n + n2 = 11

Solving for variable 'n'.

Reorder the terms:
-18 + -11 + -3n + n2 = 11 + -11

Combine like terms: -18 + -11 = -29
-29 + -3n + n2 = 11 + -11

Combine like terms: 11 + -11 = 0
-29 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '29' to each side of the equation.
-29 + -3n + 29 + n2 = 0 + 29

Reorder the terms:
-29 + 29 + -3n + n2 = 0 + 29

Combine like terms: -29 + 29 = 0
0 + -3n + n2 = 0 + 29
-3n + n2 = 0 + 29

Combine like terms: 0 + 29 = 29
-3n + n2 = 29

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 = 29 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 29 + 2.25

Combine like terms: 29 + 2.25 = 31.25
2.25 + -3n + n2 = 31.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 31.25

Calculate the square root of the right side: 5.590169944

Break this problem into two subproblems by setting 
(n + -1.5) equal to 5.590169944 and -5.590169944.

Subproblem 1

n + -1.5 = 5.590169944 Simplifying n + -1.5 = 5.590169944 Reorder the terms: -1.5 + n = 5.590169944 Solving -1.5 + n = 5.590169944 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 = 5.590169944 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 5.590169944 + 1.5 n = 5.590169944 + 1.5 Combine like terms: 5.590169944 + 1.5 = 7.090169944 n = 7.090169944 Simplifying n = 7.090169944

Subproblem 2

n + -1.5 = -5.590169944 Simplifying n + -1.5 = -5.590169944 Reorder the terms: -1.5 + n = -5.590169944 Solving -1.5 + n = -5.590169944 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 = -5.590169944 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -5.590169944 + 1.5 n = -5.590169944 + 1.5 Combine like terms: -5.590169944 + 1.5 = -4.090169944 n = -4.090169944 Simplifying n = -4.090169944

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.090169944, -4.090169944}

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