10n^2-34=65n

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Solution for 10n^2-34=65n equation:


Simplifying
10n2 + -34 = 65n

Reorder the terms:
-34 + 10n2 = 65n

Solving
-34 + 10n2 = 65n

Solving for variable 'n'.

Reorder the terms:
-34 + -65n + 10n2 = 65n + -65n

Combine like terms: 65n + -65n = 0
-34 + -65n + 10n2 = 0

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

Divide each side by '10'.
-3.4 + -6.5n + n2 = 0

Move the constant term to the right:

Add '3.4' to each side of the equation.
-3.4 + -6.5n + 3.4 + n2 = 0 + 3.4

Reorder the terms:
-3.4 + 3.4 + -6.5n + n2 = 0 + 3.4

Combine like terms: -3.4 + 3.4 = 0.0
0.0 + -6.5n + n2 = 0 + 3.4
-6.5n + n2 = 0 + 3.4

Combine like terms: 0 + 3.4 = 3.4
-6.5n + n2 = 3.4

The n term is -6.5n.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5n + 10.5625 + n2 = 3.4 + 10.5625

Reorder the terms:
10.5625 + -6.5n + n2 = 3.4 + 10.5625

Combine like terms: 3.4 + 10.5625 = 13.9625
10.5625 + -6.5n + n2 = 13.9625

Factor a perfect square on the left side:
(n + -3.25)(n + -3.25) = 13.9625

Calculate the square root of the right side: 3.736642878

Break this problem into two subproblems by setting 
(n + -3.25) equal to 3.736642878 and -3.736642878.

Subproblem 1

n + -3.25 = 3.736642878 Simplifying n + -3.25 = 3.736642878 Reorder the terms: -3.25 + n = 3.736642878 Solving -3.25 + n = 3.736642878 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + n = 3.736642878 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + n = 3.736642878 + 3.25 n = 3.736642878 + 3.25 Combine like terms: 3.736642878 + 3.25 = 6.986642878 n = 6.986642878 Simplifying n = 6.986642878

Subproblem 2

n + -3.25 = -3.736642878 Simplifying n + -3.25 = -3.736642878 Reorder the terms: -3.25 + n = -3.736642878 Solving -3.25 + n = -3.736642878 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + n = -3.736642878 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + n = -3.736642878 + 3.25 n = -3.736642878 + 3.25 Combine like terms: -3.736642878 + 3.25 = -0.486642878 n = -0.486642878 Simplifying n = -0.486642878

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.986642878, -0.486642878}

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