6n^2+7n=n^2+1

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Solution for 6n^2+7n=n^2+1 equation:


Simplifying
6n2 + 7n = n2 + 1

Reorder the terms:
7n + 6n2 = n2 + 1

Reorder the terms:
7n + 6n2 = 1 + n2

Solving
7n + 6n2 = 1 + n2

Solving for variable 'n'.

Reorder the terms:
-1 + 7n + 6n2 + -1n2 = 1 + n2 + -1 + -1n2

Combine like terms: 6n2 + -1n2 = 5n2
-1 + 7n + 5n2 = 1 + n2 + -1 + -1n2

Reorder the terms:
-1 + 7n + 5n2 = 1 + -1 + n2 + -1n2

Combine like terms: 1 + -1 = 0
-1 + 7n + 5n2 = 0 + n2 + -1n2
-1 + 7n + 5n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-1 + 7n + 5n2 = 0

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

Divide each side by '5'.
-0.2 + 1.4n + n2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 1.4n + 0.2 + n2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 1.4n + n2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 1.4n + n2 = 0 + 0.2
1.4n + n2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
1.4n + n2 = 0.2

The n term is 1.4n.  Take half its coefficient (0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
1.4n + 0.49 + n2 = 0.2 + 0.49

Reorder the terms:
0.49 + 1.4n + n2 = 0.2 + 0.49

Combine like terms: 0.2 + 0.49 = 0.69
0.49 + 1.4n + n2 = 0.69

Factor a perfect square on the left side:
(n + 0.7)(n + 0.7) = 0.69

Calculate the square root of the right side: 0.830662386

Break this problem into two subproblems by setting 
(n + 0.7) equal to 0.830662386 and -0.830662386.

Subproblem 1

n + 0.7 = 0.830662386 Simplifying n + 0.7 = 0.830662386 Reorder the terms: 0.7 + n = 0.830662386 Solving 0.7 + n = 0.830662386 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = 0.830662386 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = 0.830662386 + -0.7 n = 0.830662386 + -0.7 Combine like terms: 0.830662386 + -0.7 = 0.130662386 n = 0.130662386 Simplifying n = 0.130662386

Subproblem 2

n + 0.7 = -0.830662386 Simplifying n + 0.7 = -0.830662386 Reorder the terms: 0.7 + n = -0.830662386 Solving 0.7 + n = -0.830662386 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = -0.830662386 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = -0.830662386 + -0.7 n = -0.830662386 + -0.7 Combine like terms: -0.830662386 + -0.7 = -1.530662386 n = -1.530662386 Simplifying n = -1.530662386

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.130662386, -1.530662386}

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