3n^2=2n+3

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


Simplifying
3n2 = 2n + 3

Reorder the terms:
3n2 = 3 + 2n

Solving
3n2 = 3 + 2n

Solving for variable 'n'.

Reorder the terms:
-3 + -2n + 3n2 = 3 + 2n + -3 + -2n

Reorder the terms:
-3 + -2n + 3n2 = 3 + -3 + 2n + -2n

Combine like terms: 3 + -3 = 0
-3 + -2n + 3n2 = 0 + 2n + -2n
-3 + -2n + 3n2 = 2n + -2n

Combine like terms: 2n + -2n = 0
-3 + -2n + 3n2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-1 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.6666666667n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.6666666667n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.6666666667n + n2 = 0 + 1
-0.6666666667n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.6666666667n + n2 = 1

The n term is -0.6666666667n.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667n + 0.1111111112 + n2 = 1 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 1 + 0.1111111112

Combine like terms: 1 + 0.1111111112 = 1.1111111112
0.1111111112 + -0.6666666667n + n2 = 1.1111111112

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 1.1111111112

Calculate the square root of the right side: 1.054092553

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 1.054092553 and -1.054092553.

Subproblem 1

n + -0.3333333334 = 1.054092553 Simplifying n + -0.3333333334 = 1.054092553 Reorder the terms: -0.3333333334 + n = 1.054092553 Solving -0.3333333334 + n = 1.054092553 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = 1.054092553 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 1.054092553 + 0.3333333334 n = 1.054092553 + 0.3333333334 Combine like terms: 1.054092553 + 0.3333333334 = 1.3874258864 n = 1.3874258864 Simplifying n = 1.3874258864

Subproblem 2

n + -0.3333333334 = -1.054092553 Simplifying n + -0.3333333334 = -1.054092553 Reorder the terms: -0.3333333334 + n = -1.054092553 Solving -0.3333333334 + n = -1.054092553 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = -1.054092553 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -1.054092553 + 0.3333333334 n = -1.054092553 + 0.3333333334 Combine like terms: -1.054092553 + 0.3333333334 = -0.7207592196 n = -0.7207592196 Simplifying n = -0.7207592196

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.3874258864, -0.7207592196}

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