n^2-5n+3=8n^2+8n-4

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Solution for n^2-5n+3=8n^2+8n-4 equation:


Simplifying
n2 + -5n + 3 = 8n2 + 8n + -4

Reorder the terms:
3 + -5n + n2 = 8n2 + 8n + -4

Reorder the terms:
3 + -5n + n2 = -4 + 8n + 8n2

Solving
3 + -5n + n2 = -4 + 8n + 8n2

Solving for variable 'n'.

Reorder the terms:
3 + 4 + -5n + -8n + n2 + -8n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2

Combine like terms: 3 + 4 = 7
7 + -5n + -8n + n2 + -8n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2

Combine like terms: -5n + -8n = -13n
7 + -13n + n2 + -8n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2

Combine like terms: n2 + -8n2 = -7n2
7 + -13n + -7n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2

Reorder the terms:
7 + -13n + -7n2 = -4 + 4 + 8n + -8n + 8n2 + -8n2

Combine like terms: -4 + 4 = 0
7 + -13n + -7n2 = 0 + 8n + -8n + 8n2 + -8n2
7 + -13n + -7n2 = 8n + -8n + 8n2 + -8n2

Combine like terms: 8n + -8n = 0
7 + -13n + -7n2 = 0 + 8n2 + -8n2
7 + -13n + -7n2 = 8n2 + -8n2

Combine like terms: 8n2 + -8n2 = 0
7 + -13n + -7n2 = 0

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

Divide each side by '-7'.
-1 + 1.857142857n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.857142857n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.857142857n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.857142857n + n2 = 0 + 1
1.857142857n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.857142857n + n2 = 1

The n term is 1.857142857n.  Take half its coefficient (0.9285714285).
Square it (0.8622448978) and add it to both sides.

Add '0.8622448978' to each side of the equation.
1.857142857n + 0.8622448978 + n2 = 1 + 0.8622448978

Reorder the terms:
0.8622448978 + 1.857142857n + n2 = 1 + 0.8622448978

Combine like terms: 1 + 0.8622448978 = 1.8622448978
0.8622448978 + 1.857142857n + n2 = 1.8622448978

Factor a perfect square on the left side:
(n + 0.9285714285)(n + 0.9285714285) = 1.8622448978

Calculate the square root of the right side: 1.364640941

Break this problem into two subproblems by setting 
(n + 0.9285714285) equal to 1.364640941 and -1.364640941.

Subproblem 1

n + 0.9285714285 = 1.364640941 Simplifying n + 0.9285714285 = 1.364640941 Reorder the terms: 0.9285714285 + n = 1.364640941 Solving 0.9285714285 + n = 1.364640941 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + n = 1.364640941 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + n = 1.364640941 + -0.9285714285 n = 1.364640941 + -0.9285714285 Combine like terms: 1.364640941 + -0.9285714285 = 0.4360695125 n = 0.4360695125 Simplifying n = 0.4360695125

Subproblem 2

n + 0.9285714285 = -1.364640941 Simplifying n + 0.9285714285 = -1.364640941 Reorder the terms: 0.9285714285 + n = -1.364640941 Solving 0.9285714285 + n = -1.364640941 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + n = -1.364640941 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + n = -1.364640941 + -0.9285714285 n = -1.364640941 + -0.9285714285 Combine like terms: -1.364640941 + -0.9285714285 = -2.2932123695 n = -2.2932123695 Simplifying n = -2.2932123695

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.4360695125, -2.2932123695}

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