7n^2-8=-12n

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Solution for 7n^2-8=-12n equation:


Simplifying
7n2 + -8 = -12n

Reorder the terms:
-8 + 7n2 = -12n

Solving
-8 + 7n2 = -12n

Solving for variable 'n'.

Reorder the terms:
-8 + 12n + 7n2 = -12n + 12n

Combine like terms: -12n + 12n = 0
-8 + 12n + 7n2 = 0

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

Divide each side by '7'.
-1.142857143 + 1.714285714n + n2 = 0

Move the constant term to the right:

Add '1.142857143' to each side of the equation.
-1.142857143 + 1.714285714n + 1.142857143 + n2 = 0 + 1.142857143

Reorder the terms:
-1.142857143 + 1.142857143 + 1.714285714n + n2 = 0 + 1.142857143

Combine like terms: -1.142857143 + 1.142857143 = 0.000000000
0.000000000 + 1.714285714n + n2 = 0 + 1.142857143
1.714285714n + n2 = 0 + 1.142857143

Combine like terms: 0 + 1.142857143 = 1.142857143
1.714285714n + n2 = 1.142857143

The n term is 1.714285714n.  Take half its coefficient (0.857142857).
Square it (0.7346938773) and add it to both sides.

Add '0.7346938773' to each side of the equation.
1.714285714n + 0.7346938773 + n2 = 1.142857143 + 0.7346938773

Reorder the terms:
0.7346938773 + 1.714285714n + n2 = 1.142857143 + 0.7346938773

Combine like terms: 1.142857143 + 0.7346938773 = 1.8775510203
0.7346938773 + 1.714285714n + n2 = 1.8775510203

Factor a perfect square on the left side:
(n + 0.857142857)(n + 0.857142857) = 1.8775510203

Calculate the square root of the right side: 1.370237578

Break this problem into two subproblems by setting 
(n + 0.857142857) equal to 1.370237578 and -1.370237578.

Subproblem 1

n + 0.857142857 = 1.370237578 Simplifying n + 0.857142857 = 1.370237578 Reorder the terms: 0.857142857 + n = 1.370237578 Solving 0.857142857 + n = 1.370237578 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = 1.370237578 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = 1.370237578 + -0.857142857 n = 1.370237578 + -0.857142857 Combine like terms: 1.370237578 + -0.857142857 = 0.513094721 n = 0.513094721 Simplifying n = 0.513094721

Subproblem 2

n + 0.857142857 = -1.370237578 Simplifying n + 0.857142857 = -1.370237578 Reorder the terms: 0.857142857 + n = -1.370237578 Solving 0.857142857 + n = -1.370237578 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = -1.370237578 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = -1.370237578 + -0.857142857 n = -1.370237578 + -0.857142857 Combine like terms: -1.370237578 + -0.857142857 = -2.227380435 n = -2.227380435 Simplifying n = -2.227380435

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.513094721, -2.227380435}

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