5n^2+10n=7

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Solution for 5n^2+10n=7 equation:


Simplifying
5n2 + 10n = 7

Reorder the terms:
10n + 5n2 = 7

Solving
10n + 5n2 = 7

Solving for variable 'n'.

Reorder the terms:
-7 + 10n + 5n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 10n + 5n2 = 0

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

Divide each side by '5'.
-1.4 + 2n + n2 = 0

Move the constant term to the right:

Add '1.4' to each side of the equation.
-1.4 + 2n + 1.4 + n2 = 0 + 1.4

Reorder the terms:
-1.4 + 1.4 + 2n + n2 = 0 + 1.4

Combine like terms: -1.4 + 1.4 = 0.0
0.0 + 2n + n2 = 0 + 1.4
2n + n2 = 0 + 1.4

Combine like terms: 0 + 1.4 = 1.4
2n + n2 = 1.4

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 1.4 + 1

Reorder the terms:
1 + 2n + n2 = 1.4 + 1

Combine like terms: 1.4 + 1 = 2.4
1 + 2n + n2 = 2.4

Factor a perfect square on the left side:
(n + 1)(n + 1) = 2.4

Calculate the square root of the right side: 1.549193338

Break this problem into two subproblems by setting 
(n + 1) equal to 1.549193338 and -1.549193338.

Subproblem 1

n + 1 = 1.549193338 Simplifying n + 1 = 1.549193338 Reorder the terms: 1 + n = 1.549193338 Solving 1 + n = 1.549193338 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.549193338 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.549193338 + -1 n = 1.549193338 + -1 Combine like terms: 1.549193338 + -1 = 0.549193338 n = 0.549193338 Simplifying n = 0.549193338

Subproblem 2

n + 1 = -1.549193338 Simplifying n + 1 = -1.549193338 Reorder the terms: 1 + n = -1.549193338 Solving 1 + n = -1.549193338 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.549193338 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.549193338 + -1 n = -1.549193338 + -1 Combine like terms: -1.549193338 + -1 = -2.549193338 n = -2.549193338 Simplifying n = -2.549193338

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.549193338, -2.549193338}

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