10n^2=-3n+16

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Solution for 10n^2=-3n+16 equation:


Simplifying
10n2 = -3n + 16

Reorder the terms:
10n2 = 16 + -3n

Solving
10n2 = 16 + -3n

Solving for variable 'n'.

Reorder the terms:
-16 + 3n + 10n2 = 16 + -3n + -16 + 3n

Reorder the terms:
-16 + 3n + 10n2 = 16 + -16 + -3n + 3n

Combine like terms: 16 + -16 = 0
-16 + 3n + 10n2 = 0 + -3n + 3n
-16 + 3n + 10n2 = -3n + 3n

Combine like terms: -3n + 3n = 0
-16 + 3n + 10n2 = 0

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

Divide each side by '10'.
-1.6 + 0.3n + n2 = 0

Move the constant term to the right:

Add '1.6' to each side of the equation.
-1.6 + 0.3n + 1.6 + n2 = 0 + 1.6

Reorder the terms:
-1.6 + 1.6 + 0.3n + n2 = 0 + 1.6

Combine like terms: -1.6 + 1.6 = 0.0
0.0 + 0.3n + n2 = 0 + 1.6
0.3n + n2 = 0 + 1.6

Combine like terms: 0 + 1.6 = 1.6
0.3n + n2 = 1.6

The n term is 0.3n.  Take half its coefficient (0.15).
Square it (0.0225) and add it to both sides.

Add '0.0225' to each side of the equation.
0.3n + 0.0225 + n2 = 1.6 + 0.0225

Reorder the terms:
0.0225 + 0.3n + n2 = 1.6 + 0.0225

Combine like terms: 1.6 + 0.0225 = 1.6225
0.0225 + 0.3n + n2 = 1.6225

Factor a perfect square on the left side:
(n + 0.15)(n + 0.15) = 1.6225

Calculate the square root of the right side: 1.27377392

Break this problem into two subproblems by setting 
(n + 0.15) equal to 1.27377392 and -1.27377392.

Subproblem 1

n + 0.15 = 1.27377392 Simplifying n + 0.15 = 1.27377392 Reorder the terms: 0.15 + n = 1.27377392 Solving 0.15 + n = 1.27377392 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + n = 1.27377392 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + n = 1.27377392 + -0.15 n = 1.27377392 + -0.15 Combine like terms: 1.27377392 + -0.15 = 1.12377392 n = 1.12377392 Simplifying n = 1.12377392

Subproblem 2

n + 0.15 = -1.27377392 Simplifying n + 0.15 = -1.27377392 Reorder the terms: 0.15 + n = -1.27377392 Solving 0.15 + n = -1.27377392 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + n = -1.27377392 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + n = -1.27377392 + -0.15 n = -1.27377392 + -0.15 Combine like terms: -1.27377392 + -0.15 = -1.42377392 n = -1.42377392 Simplifying n = -1.42377392

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.12377392, -1.42377392}

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