2n^2+10n=25

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Solution for 2n^2+10n=25 equation:


Simplifying
2n2 + 10n = 25

Reorder the terms:
10n + 2n2 = 25

Solving
10n + 2n2 = 25

Solving for variable 'n'.

Reorder the terms:
-25 + 10n + 2n2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + 10n + 2n2 = 0

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

Divide each side by '2'.
-12.5 + 5n + n2 = 0

Move the constant term to the right:

Add '12.5' to each side of the equation.
-12.5 + 5n + 12.5 + n2 = 0 + 12.5

Reorder the terms:
-12.5 + 12.5 + 5n + n2 = 0 + 12.5

Combine like terms: -12.5 + 12.5 = 0.0
0.0 + 5n + n2 = 0 + 12.5
5n + n2 = 0 + 12.5

Combine like terms: 0 + 12.5 = 12.5
5n + n2 = 12.5

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 = 12.5 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 12.5 + 6.25

Combine like terms: 12.5 + 6.25 = 18.75
6.25 + 5n + n2 = 18.75

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 18.75

Calculate the square root of the right side: 4.330127019

Break this problem into two subproblems by setting 
(n + 2.5) equal to 4.330127019 and -4.330127019.

Subproblem 1

n + 2.5 = 4.330127019 Simplifying n + 2.5 = 4.330127019 Reorder the terms: 2.5 + n = 4.330127019 Solving 2.5 + n = 4.330127019 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 = 4.330127019 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 4.330127019 + -2.5 n = 4.330127019 + -2.5 Combine like terms: 4.330127019 + -2.5 = 1.830127019 n = 1.830127019 Simplifying n = 1.830127019

Subproblem 2

n + 2.5 = -4.330127019 Simplifying n + 2.5 = -4.330127019 Reorder the terms: 2.5 + n = -4.330127019 Solving 2.5 + n = -4.330127019 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 = -4.330127019 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -4.330127019 + -2.5 n = -4.330127019 + -2.5 Combine like terms: -4.330127019 + -2.5 = -6.830127019 n = -6.830127019 Simplifying n = -6.830127019

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.830127019, -6.830127019}

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