3n^2+13n-3=0

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Solution for 3n^2+13n-3=0 equation:


Simplifying
3n2 + 13n + -3 = 0

Reorder the terms:
-3 + 13n + 3n2 = 0

Solving
-3 + 13n + 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'.
-1 + 4.333333333n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 4.333333333n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 4.333333333n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 4.333333333n + n2 = 0 + 1
4.333333333n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
4.333333333n + n2 = 1

The n term is 4.333333333n.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333n + 4.694444446 + n2 = 1 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333n + n2 = 1 + 4.694444446

Combine like terms: 1 + 4.694444446 = 5.694444446
4.694444446 + 4.333333333n + n2 = 5.694444446

Factor a perfect square on the left side:
(n + 2.166666667)(n + 2.166666667) = 5.694444446

Calculate the square root of the right side: 2.386303511

Break this problem into two subproblems by setting 
(n + 2.166666667) equal to 2.386303511 and -2.386303511.

Subproblem 1

n + 2.166666667 = 2.386303511 Simplifying n + 2.166666667 = 2.386303511 Reorder the terms: 2.166666667 + n = 2.386303511 Solving 2.166666667 + n = 2.386303511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + n = 2.386303511 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + n = 2.386303511 + -2.166666667 n = 2.386303511 + -2.166666667 Combine like terms: 2.386303511 + -2.166666667 = 0.219636844 n = 0.219636844 Simplifying n = 0.219636844

Subproblem 2

n + 2.166666667 = -2.386303511 Simplifying n + 2.166666667 = -2.386303511 Reorder the terms: 2.166666667 + n = -2.386303511 Solving 2.166666667 + n = -2.386303511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + n = -2.386303511 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + n = -2.386303511 + -2.166666667 n = -2.386303511 + -2.166666667 Combine like terms: -2.386303511 + -2.166666667 = -4.552970178 n = -4.552970178 Simplifying n = -4.552970178

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.219636844, -4.552970178}

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