n^2-3n-3=0

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Solution for n^2-3n-3=0 equation:


Simplifying
n2 + -3n + -3 = 0

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

Solving
-3 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 3 + 2.25

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

Combine like terms: 3 + 2.25 = 5.25
2.25 + -3n + n2 = 5.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(n + -1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

n + -1.5 = 2.291287847 Simplifying n + -1.5 = 2.291287847 Reorder the terms: -1.5 + n = 2.291287847 Solving -1.5 + n = 2.291287847 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 = 2.291287847 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.291287847 + 1.5 n = 2.291287847 + 1.5 Combine like terms: 2.291287847 + 1.5 = 3.791287847 n = 3.791287847 Simplifying n = 3.791287847

Subproblem 2

n + -1.5 = -2.291287847 Simplifying n + -1.5 = -2.291287847 Reorder the terms: -1.5 + n = -2.291287847 Solving -1.5 + n = -2.291287847 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 = -2.291287847 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.291287847 + 1.5 n = -2.291287847 + 1.5 Combine like terms: -2.291287847 + 1.5 = -0.791287847 n = -0.791287847 Simplifying n = -0.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.791287847, -0.791287847}

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