3n^2+8n+3=0

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Solution for 3n^2+8n+3=0 equation:


Simplifying
3n2 + 8n + 3 = 0

Reorder the terms:
3 + 8n + 3n2 = 0

Solving
3 + 8n + 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 + 2.666666667n + n2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.666666667n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + 2.666666667n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.666666667n + n2 = 0 + -1
2.666666667n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.666666667n + n2 = -1

The n term is 2.666666667n.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667n + 1.777777780 + n2 = -1 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667n + n2 = -1 + 1.777777780

Combine like terms: -1 + 1.777777780 = 0.77777778
1.777777780 + 2.666666667n + n2 = 0.77777778

Factor a perfect square on the left side:
(n + 1.333333334)(n + 1.333333334) = 0.77777778

Calculate the square root of the right side: 0.881917105

Break this problem into two subproblems by setting 
(n + 1.333333334) equal to 0.881917105 and -0.881917105.

Subproblem 1

n + 1.333333334 = 0.881917105 Simplifying n + 1.333333334 = 0.881917105 Reorder the terms: 1.333333334 + n = 0.881917105 Solving 1.333333334 + n = 0.881917105 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = 0.881917105 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = 0.881917105 + -1.333333334 n = 0.881917105 + -1.333333334 Combine like terms: 0.881917105 + -1.333333334 = -0.451416229 n = -0.451416229 Simplifying n = -0.451416229

Subproblem 2

n + 1.333333334 = -0.881917105 Simplifying n + 1.333333334 = -0.881917105 Reorder the terms: 1.333333334 + n = -0.881917105 Solving 1.333333334 + n = -0.881917105 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = -0.881917105 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = -0.881917105 + -1.333333334 n = -0.881917105 + -1.333333334 Combine like terms: -0.881917105 + -1.333333334 = -2.215250439 n = -2.215250439 Simplifying n = -2.215250439

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.451416229, -2.215250439}

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