n^2+6n=10

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Solution for n^2+6n=10 equation:


Simplifying
n2 + 6n = 10

Reorder the terms:
6n + n2 = 10

Solving
6n + n2 = 10

Solving for variable 'n'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 6n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + 6n + n2 = 0 + 10

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

Combine like terms: 0 + 10 = 10
6n + n2 = 10

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = 10 + 9

Reorder the terms:
9 + 6n + n2 = 10 + 9

Combine like terms: 10 + 9 = 19
9 + 6n + n2 = 19

Factor a perfect square on the left side:
(n + 3)(n + 3) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(n + 3) equal to 4.358898944 and -4.358898944.

Subproblem 1

n + 3 = 4.358898944 Simplifying n + 3 = 4.358898944 Reorder the terms: 3 + n = 4.358898944 Solving 3 + n = 4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 4.358898944 + -3 Combine like terms: 3 + -3 = 0 0 + n = 4.358898944 + -3 n = 4.358898944 + -3 Combine like terms: 4.358898944 + -3 = 1.358898944 n = 1.358898944 Simplifying n = 1.358898944

Subproblem 2

n + 3 = -4.358898944 Simplifying n + 3 = -4.358898944 Reorder the terms: 3 + n = -4.358898944 Solving 3 + n = -4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -4.358898944 + -3 Combine like terms: 3 + -3 = 0 0 + n = -4.358898944 + -3 n = -4.358898944 + -3 Combine like terms: -4.358898944 + -3 = -7.358898944 n = -7.358898944 Simplifying n = -7.358898944

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.358898944, -7.358898944}

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