n^2+7n=10

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


Simplifying
n2 + 7n = 10

Reorder the terms:
7n + n2 = 10

Solving
7n + n2 = 10

Solving for variable 'n'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 7n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + 7n + n2 = 0 + 10

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

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

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 10 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 10 + 12.25

Combine like terms: 10 + 12.25 = 22.25
12.25 + 7n + n2 = 22.25

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(n + 3.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

n + 3.5 = 4.716990566 Simplifying n + 3.5 = 4.716990566 Reorder the terms: 3.5 + n = 4.716990566 Solving 3.5 + n = 4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 4.716990566 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.716990566 + -3.5 n = 4.716990566 + -3.5 Combine like terms: 4.716990566 + -3.5 = 1.216990566 n = 1.216990566 Simplifying n = 1.216990566

Subproblem 2

n + 3.5 = -4.716990566 Simplifying n + 3.5 = -4.716990566 Reorder the terms: 3.5 + n = -4.716990566 Solving 3.5 + n = -4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -4.716990566 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.716990566 + -3.5 n = -4.716990566 + -3.5 Combine like terms: -4.716990566 + -3.5 = -8.216990566 n = -8.216990566 Simplifying n = -8.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.216990566, -8.216990566}

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