10n^2-5n-10=-6

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Solution for 10n^2-5n-10=-6 equation:


Simplifying
10n2 + -5n + -10 = -6

Reorder the terms:
-10 + -5n + 10n2 = -6

Solving
-10 + -5n + 10n2 = -6

Solving for variable 'n'.

Reorder the terms:
-10 + 6 + -5n + 10n2 = -6 + 6

Combine like terms: -10 + 6 = -4
-4 + -5n + 10n2 = -6 + 6

Combine like terms: -6 + 6 = 0
-4 + -5n + 10n2 = 0

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

Divide each side by '10'.
-0.4 + -0.5n + n2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + -0.5n + 0.4 + n2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + -0.5n + n2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + -0.5n + n2 = 0 + 0.4
-0.5n + n2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
-0.5n + n2 = 0.4

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 = 0.4 + 0.0625

Reorder the terms:
0.0625 + -0.5n + n2 = 0.4 + 0.0625

Combine like terms: 0.4 + 0.0625 = 0.4625
0.0625 + -0.5n + n2 = 0.4625

Factor a perfect square on the left side:
(n + -0.25)(n + -0.25) = 0.4625

Calculate the square root of the right side: 0.680073525

Break this problem into two subproblems by setting 
(n + -0.25) equal to 0.680073525 and -0.680073525.

Subproblem 1

n + -0.25 = 0.680073525 Simplifying n + -0.25 = 0.680073525 Reorder the terms: -0.25 + n = 0.680073525 Solving -0.25 + n = 0.680073525 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 = 0.680073525 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 0.680073525 + 0.25 n = 0.680073525 + 0.25 Combine like terms: 0.680073525 + 0.25 = 0.930073525 n = 0.930073525 Simplifying n = 0.930073525

Subproblem 2

n + -0.25 = -0.680073525 Simplifying n + -0.25 = -0.680073525 Reorder the terms: -0.25 + n = -0.680073525 Solving -0.25 + n = -0.680073525 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 = -0.680073525 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -0.680073525 + 0.25 n = -0.680073525 + 0.25 Combine like terms: -0.680073525 + 0.25 = -0.430073525 n = -0.430073525 Simplifying n = -0.430073525

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.930073525, -0.430073525}

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