n(7n+12)=20

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Solution for n(7n+12)=20 equation:


Simplifying
n(7n + 12) = 20

Reorder the terms:
n(12 + 7n) = 20
(12 * n + 7n * n) = 20
(12n + 7n2) = 20

Solving
12n + 7n2 = 20

Solving for variable 'n'.

Reorder the terms:
-20 + 12n + 7n2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + 12n + 7n2 = 0

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

Divide each side by '7'.
-2.857142857 + 1.714285714n + n2 = 0

Move the constant term to the right:

Add '2.857142857' to each side of the equation.
-2.857142857 + 1.714285714n + 2.857142857 + n2 = 0 + 2.857142857

Reorder the terms:
-2.857142857 + 2.857142857 + 1.714285714n + n2 = 0 + 2.857142857

Combine like terms: -2.857142857 + 2.857142857 = 0.000000000
0.000000000 + 1.714285714n + n2 = 0 + 2.857142857
1.714285714n + n2 = 0 + 2.857142857

Combine like terms: 0 + 2.857142857 = 2.857142857
1.714285714n + n2 = 2.857142857

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 = 2.857142857 + 0.7346938773

Reorder the terms:
0.7346938773 + 1.714285714n + n2 = 2.857142857 + 0.7346938773

Combine like terms: 2.857142857 + 0.7346938773 = 3.5918367343
0.7346938773 + 1.714285714n + n2 = 3.5918367343

Factor a perfect square on the left side:
(n + 0.857142857)(n + 0.857142857) = 3.5918367343

Calculate the square root of the right side: 1.895214166

Break this problem into two subproblems by setting 
(n + 0.857142857) equal to 1.895214166 and -1.895214166.

Subproblem 1

n + 0.857142857 = 1.895214166 Simplifying n + 0.857142857 = 1.895214166 Reorder the terms: 0.857142857 + n = 1.895214166 Solving 0.857142857 + n = 1.895214166 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 = 1.895214166 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = 1.895214166 + -0.857142857 n = 1.895214166 + -0.857142857 Combine like terms: 1.895214166 + -0.857142857 = 1.038071309 n = 1.038071309 Simplifying n = 1.038071309

Subproblem 2

n + 0.857142857 = -1.895214166 Simplifying n + 0.857142857 = -1.895214166 Reorder the terms: 0.857142857 + n = -1.895214166 Solving 0.857142857 + n = -1.895214166 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 = -1.895214166 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = -1.895214166 + -0.857142857 n = -1.895214166 + -0.857142857 Combine like terms: -1.895214166 + -0.857142857 = -2.752357023 n = -2.752357023 Simplifying n = -2.752357023

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.038071309, -2.752357023}

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