3n^2-7n-18=0

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Solution for 3n^2-7n-18=0 equation:


Simplifying
3n2 + -7n + -18 = 0

Reorder the terms:
-18 + -7n + 3n2 = 0

Solving
-18 + -7n + 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'.
-6 + -2.333333333n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -2.333333333n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -2.333333333n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -2.333333333n + n2 = 0 + 6
-2.333333333n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-2.333333333n + n2 = 6

The n term is -2.333333333n.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333n + 1.361111112 + n2 = 6 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333n + n2 = 6 + 1.361111112

Combine like terms: 6 + 1.361111112 = 7.361111112
1.361111112 + -2.333333333n + n2 = 7.361111112

Factor a perfect square on the left side:
(n + -1.166666667)(n + -1.166666667) = 7.361111112

Calculate the square root of the right side: 2.713136766

Break this problem into two subproblems by setting 
(n + -1.166666667) equal to 2.713136766 and -2.713136766.

Subproblem 1

n + -1.166666667 = 2.713136766 Simplifying n + -1.166666667 = 2.713136766 Reorder the terms: -1.166666667 + n = 2.713136766 Solving -1.166666667 + n = 2.713136766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + n = 2.713136766 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n = 2.713136766 + 1.166666667 n = 2.713136766 + 1.166666667 Combine like terms: 2.713136766 + 1.166666667 = 3.879803433 n = 3.879803433 Simplifying n = 3.879803433

Subproblem 2

n + -1.166666667 = -2.713136766 Simplifying n + -1.166666667 = -2.713136766 Reorder the terms: -1.166666667 + n = -2.713136766 Solving -1.166666667 + n = -2.713136766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + n = -2.713136766 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n = -2.713136766 + 1.166666667 n = -2.713136766 + 1.166666667 Combine like terms: -2.713136766 + 1.166666667 = -1.546470099 n = -1.546470099 Simplifying n = -1.546470099

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.879803433, -1.546470099}

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