20n+20=7n^2-n

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Solution for 20n+20=7n^2-n equation:


Simplifying
20n + 20 = 7n2 + -1n

Reorder the terms:
20 + 20n = 7n2 + -1n

Reorder the terms:
20 + 20n = -1n + 7n2

Solving
20 + 20n = -1n + 7n2

Solving for variable 'n'.

Combine like terms: 20n + n = 21n
20 + 21n + -7n2 = -1n + 7n2 + n + -7n2

Reorder the terms:
20 + 21n + -7n2 = -1n + n + 7n2 + -7n2

Combine like terms: -1n + n = 0
20 + 21n + -7n2 = 0 + 7n2 + -7n2
20 + 21n + -7n2 = 7n2 + -7n2

Combine like terms: 7n2 + -7n2 = 0
20 + 21n + -7n2 = 0

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

Divide each side by '-7'.
-2.857142857 + -3n + n2 = 0

Move the constant term to the right:

Add '2.857142857' to each side of the equation.
-2.857142857 + -3n + 2.857142857 + n2 = 0 + 2.857142857

Reorder the terms:
-2.857142857 + 2.857142857 + -3n + n2 = 0 + 2.857142857

Combine like terms: -2.857142857 + 2.857142857 = 0.000000000
0.000000000 + -3n + n2 = 0 + 2.857142857
-3n + n2 = 0 + 2.857142857

Combine like terms: 0 + 2.857142857 = 2.857142857
-3n + n2 = 2.857142857

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 2.857142857 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 2.857142857 + 2.25

Combine like terms: 2.857142857 + 2.25 = 5.107142857
2.25 + -3n + n2 = 5.107142857

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 5.107142857

Calculate the square root of the right side: 2.25989886

Break this problem into two subproblems by setting 
(n + -1.5) equal to 2.25989886 and -2.25989886.

Subproblem 1

n + -1.5 = 2.25989886 Simplifying n + -1.5 = 2.25989886 Reorder the terms: -1.5 + n = 2.25989886 Solving -1.5 + n = 2.25989886 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 2.25989886 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.25989886 + 1.5 n = 2.25989886 + 1.5 Combine like terms: 2.25989886 + 1.5 = 3.75989886 n = 3.75989886 Simplifying n = 3.75989886

Subproblem 2

n + -1.5 = -2.25989886 Simplifying n + -1.5 = -2.25989886 Reorder the terms: -1.5 + n = -2.25989886 Solving -1.5 + n = -2.25989886 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -2.25989886 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.25989886 + 1.5 n = -2.25989886 + 1.5 Combine like terms: -2.25989886 + 1.5 = -0.75989886 n = -0.75989886 Simplifying n = -0.75989886

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.75989886, -0.75989886}

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