2n^2=3n+12

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Solution for 2n^2=3n+12 equation:


Simplifying
2n2 = 3n + 12

Reorder the terms:
2n2 = 12 + 3n

Solving
2n2 = 12 + 3n

Solving for variable 'n'.

Reorder the terms:
-12 + -3n + 2n2 = 12 + 3n + -12 + -3n

Reorder the terms:
-12 + -3n + 2n2 = 12 + -12 + 3n + -3n

Combine like terms: 12 + -12 = 0
-12 + -3n + 2n2 = 0 + 3n + -3n
-12 + -3n + 2n2 = 3n + -3n

Combine like terms: 3n + -3n = 0
-12 + -3n + 2n2 = 0

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

Divide each side by '2'.
-6 + -1.5n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -1.5n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -1.5n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -1.5n + n2 = 0 + 6
-1.5n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-1.5n + n2 = 6

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = 6 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 6 + 0.5625

Combine like terms: 6 + 0.5625 = 6.5625
0.5625 + -1.5n + n2 = 6.5625

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(n + -0.75) equal to 2.561737691 and -2.561737691.

Subproblem 1

n + -0.75 = 2.561737691 Simplifying n + -0.75 = 2.561737691 Reorder the terms: -0.75 + n = 2.561737691 Solving -0.75 + n = 2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 2.561737691 + 0.75 n = 2.561737691 + 0.75 Combine like terms: 2.561737691 + 0.75 = 3.311737691 n = 3.311737691 Simplifying n = 3.311737691

Subproblem 2

n + -0.75 = -2.561737691 Simplifying n + -0.75 = -2.561737691 Reorder the terms: -0.75 + n = -2.561737691 Solving -0.75 + n = -2.561737691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -2.561737691 + 0.75 n = -2.561737691 + 0.75 Combine like terms: -2.561737691 + 0.75 = -1.811737691 n = -1.811737691 Simplifying n = -1.811737691

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.311737691, -1.811737691}

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