n^2-3n-34=0

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


Simplifying
n2 + -3n + -34 = 0

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

Solving
-34 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + -3n + 34 + n2 = 0 + 34

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

Combine like terms: -34 + 34 = 0
0 + -3n + n2 = 0 + 34
-3n + n2 = 0 + 34

Combine like terms: 0 + 34 = 34
-3n + n2 = 34

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 34 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 34 + 2.25

Combine like terms: 34 + 2.25 = 36.25
2.25 + -3n + n2 = 36.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 36.25

Calculate the square root of the right side: 6.020797289

Break this problem into two subproblems by setting 
(n + -1.5) equal to 6.020797289 and -6.020797289.

Subproblem 1

n + -1.5 = 6.020797289 Simplifying n + -1.5 = 6.020797289 Reorder the terms: -1.5 + n = 6.020797289 Solving -1.5 + n = 6.020797289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 6.020797289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 6.020797289 + 1.5 n = 6.020797289 + 1.5 Combine like terms: 6.020797289 + 1.5 = 7.520797289 n = 7.520797289 Simplifying n = 7.520797289

Subproblem 2

n + -1.5 = -6.020797289 Simplifying n + -1.5 = -6.020797289 Reorder the terms: -1.5 + n = -6.020797289 Solving -1.5 + n = -6.020797289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -6.020797289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -6.020797289 + 1.5 n = -6.020797289 + 1.5 Combine like terms: -6.020797289 + 1.5 = -4.520797289 n = -4.520797289 Simplifying n = -4.520797289

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.520797289, -4.520797289}

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