7n^2-38n-34=0

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Solution for 7n^2-38n-34=0 equation:


Simplifying
7n2 + -38n + -34 = 0

Reorder the terms:
-34 + -38n + 7n2 = 0

Solving
-34 + -38n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
-4.857142857 + -5.428571429n + n2 = 0

Move the constant term to the right:

Add '4.857142857' to each side of the equation.
-4.857142857 + -5.428571429n + 4.857142857 + n2 = 0 + 4.857142857

Reorder the terms:
-4.857142857 + 4.857142857 + -5.428571429n + n2 = 0 + 4.857142857

Combine like terms: -4.857142857 + 4.857142857 = 0.000000000
0.000000000 + -5.428571429n + n2 = 0 + 4.857142857
-5.428571429n + n2 = 0 + 4.857142857

Combine like terms: 0 + 4.857142857 = 4.857142857
-5.428571429n + n2 = 4.857142857

The n term is -5.428571429n.  Take half its coefficient (-2.714285715).
Square it (7.367346943) and add it to both sides.

Add '7.367346943' to each side of the equation.
-5.428571429n + 7.367346943 + n2 = 4.857142857 + 7.367346943

Reorder the terms:
7.367346943 + -5.428571429n + n2 = 4.857142857 + 7.367346943

Combine like terms: 4.857142857 + 7.367346943 = 12.2244898
7.367346943 + -5.428571429n + n2 = 12.2244898

Factor a perfect square on the left side:
(n + -2.714285715)(n + -2.714285715) = 12.2244898

Calculate the square root of the right side: 3.496353786

Break this problem into two subproblems by setting 
(n + -2.714285715) equal to 3.496353786 and -3.496353786.

Subproblem 1

n + -2.714285715 = 3.496353786 Simplifying n + -2.714285715 = 3.496353786 Reorder the terms: -2.714285715 + n = 3.496353786 Solving -2.714285715 + n = 3.496353786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.714285715' to each side of the equation. -2.714285715 + 2.714285715 + n = 3.496353786 + 2.714285715 Combine like terms: -2.714285715 + 2.714285715 = 0.000000000 0.000000000 + n = 3.496353786 + 2.714285715 n = 3.496353786 + 2.714285715 Combine like terms: 3.496353786 + 2.714285715 = 6.210639501 n = 6.210639501 Simplifying n = 6.210639501

Subproblem 2

n + -2.714285715 = -3.496353786 Simplifying n + -2.714285715 = -3.496353786 Reorder the terms: -2.714285715 + n = -3.496353786 Solving -2.714285715 + n = -3.496353786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.714285715' to each side of the equation. -2.714285715 + 2.714285715 + n = -3.496353786 + 2.714285715 Combine like terms: -2.714285715 + 2.714285715 = 0.000000000 0.000000000 + n = -3.496353786 + 2.714285715 n = -3.496353786 + 2.714285715 Combine like terms: -3.496353786 + 2.714285715 = -0.782068071 n = -0.782068071 Simplifying n = -0.782068071

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.210639501, -0.782068071}

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