10n^2-9=5n

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Solution for 10n^2-9=5n equation:


Simplifying
10n2 + -9 = 5n

Reorder the terms:
-9 + 10n2 = 5n

Solving
-9 + 10n2 = 5n

Solving for variable 'n'.

Reorder the terms:
-9 + -5n + 10n2 = 5n + -5n

Combine like terms: 5n + -5n = 0
-9 + -5n + 10n2 = 0

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

Divide each side by '10'.
-0.9 + -0.5n + n2 = 0

Move the constant term to the right:

Add '0.9' to each side of the equation.
-0.9 + -0.5n + 0.9 + n2 = 0 + 0.9

Reorder the terms:
-0.9 + 0.9 + -0.5n + n2 = 0 + 0.9

Combine like terms: -0.9 + 0.9 = 0.0
0.0 + -0.5n + n2 = 0 + 0.9
-0.5n + n2 = 0 + 0.9

Combine like terms: 0 + 0.9 = 0.9
-0.5n + n2 = 0.9

The n term is -0.5n.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5n + 0.0625 + n2 = 0.9 + 0.0625

Reorder the terms:
0.0625 + -0.5n + n2 = 0.9 + 0.0625

Combine like terms: 0.9 + 0.0625 = 0.9625
0.0625 + -0.5n + n2 = 0.9625

Factor a perfect square on the left side:
(n + -0.25)(n + -0.25) = 0.9625

Calculate the square root of the right side: 0.981070844

Break this problem into two subproblems by setting 
(n + -0.25) equal to 0.981070844 and -0.981070844.

Subproblem 1

n + -0.25 = 0.981070844 Simplifying n + -0.25 = 0.981070844 Reorder the terms: -0.25 + n = 0.981070844 Solving -0.25 + n = 0.981070844 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = 0.981070844 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 0.981070844 + 0.25 n = 0.981070844 + 0.25 Combine like terms: 0.981070844 + 0.25 = 1.231070844 n = 1.231070844 Simplifying n = 1.231070844

Subproblem 2

n + -0.25 = -0.981070844 Simplifying n + -0.25 = -0.981070844 Reorder the terms: -0.25 + n = -0.981070844 Solving -0.25 + n = -0.981070844 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = -0.981070844 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -0.981070844 + 0.25 n = -0.981070844 + 0.25 Combine like terms: -0.981070844 + 0.25 = -0.731070844 n = -0.731070844 Simplifying n = -0.731070844

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.231070844, -0.731070844}

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