3n^2-5n-9=0

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Solution for 3n^2-5n-9=0 equation:


Simplifying
3n2 + -5n + -9 = 0

Reorder the terms:
-9 + -5n + 3n2 = 0

Solving
-9 + -5n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-3 + -1.666666667n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1.666666667n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + -1.666666667n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1.666666667n + n2 = 0 + 3
-1.666666667n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1.666666667n + n2 = 3

The n term is -1.666666667n.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667n + 0.6944444447 + n2 = 3 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667n + n2 = 3 + 0.6944444447

Combine like terms: 3 + 0.6944444447 = 3.6944444447
0.6944444447 + -1.666666667n + n2 = 3.6944444447

Factor a perfect square on the left side:
(n + -0.8333333335)(n + -0.8333333335) = 3.6944444447

Calculate the square root of the right side: 1.922093766

Break this problem into two subproblems by setting 
(n + -0.8333333335) equal to 1.922093766 and -1.922093766.

Subproblem 1

n + -0.8333333335 = 1.922093766 Simplifying n + -0.8333333335 = 1.922093766 Reorder the terms: -0.8333333335 + n = 1.922093766 Solving -0.8333333335 + n = 1.922093766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = 1.922093766 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = 1.922093766 + 0.8333333335 n = 1.922093766 + 0.8333333335 Combine like terms: 1.922093766 + 0.8333333335 = 2.7554270995 n = 2.7554270995 Simplifying n = 2.7554270995

Subproblem 2

n + -0.8333333335 = -1.922093766 Simplifying n + -0.8333333335 = -1.922093766 Reorder the terms: -0.8333333335 + n = -1.922093766 Solving -0.8333333335 + n = -1.922093766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = -1.922093766 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = -1.922093766 + 0.8333333335 n = -1.922093766 + 0.8333333335 Combine like terms: -1.922093766 + 0.8333333335 = -1.0887604325 n = -1.0887604325 Simplifying n = -1.0887604325

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.7554270995, -1.0887604325}

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