3n^2-7n-3=0

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


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

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

Solving
-3 + -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'.
-1 + -2.333333333n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.333333333n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.333333333n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.333333333n + n2 = 0 + 1
-2.333333333n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.333333333n + n2 = 1

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 = 1 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333n + n2 = 1 + 1.361111112

Combine like terms: 1 + 1.361111112 = 2.361111112
1.361111112 + -2.333333333n + n2 = 2.361111112

Factor a perfect square on the left side:
(n + -1.166666667)(n + -1.166666667) = 2.361111112

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(n + -1.166666667) equal to 1.536590743 and -1.536590743.

Subproblem 1

n + -1.166666667 = 1.536590743 Simplifying n + -1.166666667 = 1.536590743 Reorder the terms: -1.166666667 + n = 1.536590743 Solving -1.166666667 + n = 1.536590743 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 = 1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n = 1.536590743 + 1.166666667 n = 1.536590743 + 1.166666667 Combine like terms: 1.536590743 + 1.166666667 = 2.70325741 n = 2.70325741 Simplifying n = 2.70325741

Subproblem 2

n + -1.166666667 = -1.536590743 Simplifying n + -1.166666667 = -1.536590743 Reorder the terms: -1.166666667 + n = -1.536590743 Solving -1.166666667 + n = -1.536590743 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 = -1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n = -1.536590743 + 1.166666667 n = -1.536590743 + 1.166666667 Combine like terms: -1.536590743 + 1.166666667 = -0.369924076 n = -0.369924076 Simplifying n = -0.369924076

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.70325741, -0.369924076}

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