3n^2=8n-3

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Solution for 3n^2=8n-3 equation:


Simplifying
3n2 = 8n + -3

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

Solving
3n2 = -3 + 8n

Solving for variable 'n'.

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

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

Combine like terms: -3 + 3 = 0
3 + -8n + 3n2 = 0 + 8n + -8n
3 + -8n + 3n2 = 8n + -8n

Combine like terms: 8n + -8n = 0
3 + -8n + 3n2 = 0

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 = 2.215250439 n = 2.215250439 Simplifying n = 2.215250439

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 = 0.451416229 n = 0.451416229 Simplifying n = 0.451416229

Solution

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

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