3n^2+21n-34=0

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Solution for 3n^2+21n-34=0 equation:


Simplifying
3n2 + 21n + -34 = 0

Reorder the terms:
-34 + 21n + 3n2 = 0

Solving
-34 + 21n + 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'.
-11.33333333 + 7n + n2 = 0

Move the constant term to the right:

Add '11.33333333' to each side of the equation.
-11.33333333 + 7n + 11.33333333 + n2 = 0 + 11.33333333

Reorder the terms:
-11.33333333 + 11.33333333 + 7n + n2 = 0 + 11.33333333

Combine like terms: -11.33333333 + 11.33333333 = 0.00000000
0.00000000 + 7n + n2 = 0 + 11.33333333
7n + n2 = 0 + 11.33333333

Combine like terms: 0 + 11.33333333 = 11.33333333
7n + n2 = 11.33333333

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 11.33333333 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 11.33333333 + 12.25

Combine like terms: 11.33333333 + 12.25 = 23.58333333
12.25 + 7n + n2 = 23.58333333

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 23.58333333

Calculate the square root of the right side: 4.856267428

Break this problem into two subproblems by setting 
(n + 3.5) equal to 4.856267428 and -4.856267428.

Subproblem 1

n + 3.5 = 4.856267428 Simplifying n + 3.5 = 4.856267428 Reorder the terms: 3.5 + n = 4.856267428 Solving 3.5 + n = 4.856267428 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 4.856267428 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.856267428 + -3.5 n = 4.856267428 + -3.5 Combine like terms: 4.856267428 + -3.5 = 1.356267428 n = 1.356267428 Simplifying n = 1.356267428

Subproblem 2

n + 3.5 = -4.856267428 Simplifying n + 3.5 = -4.856267428 Reorder the terms: 3.5 + n = -4.856267428 Solving 3.5 + n = -4.856267428 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -4.856267428 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.856267428 + -3.5 n = -4.856267428 + -3.5 Combine like terms: -4.856267428 + -3.5 = -8.356267428 n = -8.356267428 Simplifying n = -8.356267428

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.356267428, -8.356267428}

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