3a+1=3.6a(a-1)

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Solution for 3a+1=3.6a(a-1) equation:


Simplifying
3a + 1 = 3.6a(a + -1)

Reorder the terms:
1 + 3a = 3.6a(a + -1)

Reorder the terms:
1 + 3a = 3.6a(-1 + a)
1 + 3a = (-1 * 3.6a + a * 3.6a)
1 + 3a = (-3.6a + 3.6a2)

Solving
1 + 3a = -3.6a + 3.6a2

Solving for variable 'a'.

Combine like terms: 3a + 3.6a = 6.6a
1 + 6.6a + -3.6a2 = -3.6a + 3.6a2 + 3.6a + -3.6a2

Reorder the terms:
1 + 6.6a + -3.6a2 = -3.6a + 3.6a + 3.6a2 + -3.6a2

Combine like terms: -3.6a + 3.6a = 0.0
1 + 6.6a + -3.6a2 = 0.0 + 3.6a2 + -3.6a2
1 + 6.6a + -3.6a2 = 3.6a2 + -3.6a2

Combine like terms: 3.6a2 + -3.6a2 = 0.0
1 + 6.6a + -3.6a2 = 0.0

Begin completing the square.  Divide all terms by
-3.6 the coefficient of the squared term: 

Divide each side by '-3.6'.
-0.2777777778 + -1.833333333a + a2 = 0

Move the constant term to the right:

Add '0.2777777778' to each side of the equation.
-0.2777777778 + -1.833333333a + 0.2777777778 + a2 = 0 + 0.2777777778

Reorder the terms:
-0.2777777778 + 0.2777777778 + -1.833333333a + a2 = 0 + 0.2777777778

Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000
0.0000000000 + -1.833333333a + a2 = 0 + 0.2777777778
-1.833333333a + a2 = 0 + 0.2777777778

Combine like terms: 0 + 0.2777777778 = 0.2777777778
-1.833333333a + a2 = 0.2777777778

The a term is -1.833333333a.  Take half its coefficient (-0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
-1.833333333a + 0.8402777775 + a2 = 0.2777777778 + 0.8402777775

Reorder the terms:
0.8402777775 + -1.833333333a + a2 = 0.2777777778 + 0.8402777775

Combine like terms: 0.2777777778 + 0.8402777775 = 1.1180555553
0.8402777775 + -1.833333333a + a2 = 1.1180555553

Factor a perfect square on the left side:
(a + -0.9166666665)(a + -0.9166666665) = 1.1180555553

Calculate the square root of the right side: 1.057381462

Break this problem into two subproblems by setting 
(a + -0.9166666665) equal to 1.057381462 and -1.057381462.

Subproblem 1

a + -0.9166666665 = 1.057381462 Simplifying a + -0.9166666665 = 1.057381462 Reorder the terms: -0.9166666665 + a = 1.057381462 Solving -0.9166666665 + a = 1.057381462 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + a = 1.057381462 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + a = 1.057381462 + 0.9166666665 a = 1.057381462 + 0.9166666665 Combine like terms: 1.057381462 + 0.9166666665 = 1.9740481285 a = 1.9740481285 Simplifying a = 1.9740481285

Subproblem 2

a + -0.9166666665 = -1.057381462 Simplifying a + -0.9166666665 = -1.057381462 Reorder the terms: -0.9166666665 + a = -1.057381462 Solving -0.9166666665 + a = -1.057381462 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + a = -1.057381462 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + a = -1.057381462 + 0.9166666665 a = -1.057381462 + 0.9166666665 Combine like terms: -1.057381462 + 0.9166666665 = -0.1407147955 a = -0.1407147955 Simplifying a = -0.1407147955

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.9740481285, -0.1407147955}

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