2n^2+5n-33=12

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Solution for 2n^2+5n-33=12 equation:


Simplifying
2n2 + 5n + -33 = 12

Reorder the terms:
-33 + 5n + 2n2 = 12

Solving
-33 + 5n + 2n2 = 12

Solving for variable 'n'.

Reorder the terms:
-33 + -12 + 5n + 2n2 = 12 + -12

Combine like terms: -33 + -12 = -45
-45 + 5n + 2n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-45 + 5n + 2n2 = 0

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

Divide each side by '2'.
-22.5 + 2.5n + n2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 2.5n + 22.5 + n2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 2.5n + n2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 2.5n + n2 = 0 + 22.5
2.5n + n2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
2.5n + n2 = 22.5

The n term is 2.5n.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5n + 1.5625 + n2 = 22.5 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = 22.5 + 1.5625

Combine like terms: 22.5 + 1.5625 = 24.0625
1.5625 + 2.5n + n2 = 24.0625

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 24.0625

Calculate the square root of the right side: 4.905354218

Break this problem into two subproblems by setting 
(n + 1.25) equal to 4.905354218 and -4.905354218.

Subproblem 1

n + 1.25 = 4.905354218 Simplifying n + 1.25 = 4.905354218 Reorder the terms: 1.25 + n = 4.905354218 Solving 1.25 + n = 4.905354218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 4.905354218 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 4.905354218 + -1.25 n = 4.905354218 + -1.25 Combine like terms: 4.905354218 + -1.25 = 3.655354218 n = 3.655354218 Simplifying n = 3.655354218

Subproblem 2

n + 1.25 = -4.905354218 Simplifying n + 1.25 = -4.905354218 Reorder the terms: 1.25 + n = -4.905354218 Solving 1.25 + n = -4.905354218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -4.905354218 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -4.905354218 + -1.25 n = -4.905354218 + -1.25 Combine like terms: -4.905354218 + -1.25 = -6.155354218 n = -6.155354218 Simplifying n = -6.155354218

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.655354218, -6.155354218}

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