2n^2=-10n-3

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


Simplifying
2n2 = -10n + -3

Reorder the terms:
2n2 = -3 + -10n

Solving
2n2 = -3 + -10n

Solving for variable 'n'.

Reorder the terms:
3 + 10n + 2n2 = -3 + -10n + 3 + 10n

Reorder the terms:
3 + 10n + 2n2 = -3 + 3 + -10n + 10n

Combine like terms: -3 + 3 = 0
3 + 10n + 2n2 = 0 + -10n + 10n
3 + 10n + 2n2 = -10n + 10n

Combine like terms: -10n + 10n = 0
3 + 10n + 2n2 = 0

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

Divide each side by '2'.
1.5 + 5n + n2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 5n + -1.5 + n2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 5n + n2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 5n + n2 = 0 + -1.5
5n + n2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
5n + n2 = -1.5

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = -1.5 + 6.25

Reorder the terms:
6.25 + 5n + n2 = -1.5 + 6.25

Combine like terms: -1.5 + 6.25 = 4.75
6.25 + 5n + n2 = 4.75

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 4.75

Calculate the square root of the right side: 2.179449472

Break this problem into two subproblems by setting 
(n + 2.5) equal to 2.179449472 and -2.179449472.

Subproblem 1

n + 2.5 = 2.179449472 Simplifying n + 2.5 = 2.179449472 Reorder the terms: 2.5 + n = 2.179449472 Solving 2.5 + n = 2.179449472 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 2.179449472 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 2.179449472 + -2.5 n = 2.179449472 + -2.5 Combine like terms: 2.179449472 + -2.5 = -0.320550528 n = -0.320550528 Simplifying n = -0.320550528

Subproblem 2

n + 2.5 = -2.179449472 Simplifying n + 2.5 = -2.179449472 Reorder the terms: 2.5 + n = -2.179449472 Solving 2.5 + n = -2.179449472 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -2.179449472 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -2.179449472 + -2.5 n = -2.179449472 + -2.5 Combine like terms: -2.179449472 + -2.5 = -4.679449472 n = -4.679449472 Simplifying n = -4.679449472

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.320550528, -4.679449472}

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