(3a^2-1)-6a=5a

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Solution for (3a^2-1)-6a=5a equation:


Simplifying
(3a2 + -1) + -6a = 5a

Reorder the terms:
(-1 + 3a2) + -6a = 5a

Remove parenthesis around (-1 + 3a2)
-1 + 3a2 + -6a = 5a

Reorder the terms:
-1 + -6a + 3a2 = 5a

Solving
-1 + -6a + 3a2 = 5a

Solving for variable 'a'.

Reorder the terms:
-1 + -6a + -5a + 3a2 = 5a + -5a

Combine like terms: -6a + -5a = -11a
-1 + -11a + 3a2 = 5a + -5a

Combine like terms: 5a + -5a = 0
-1 + -11a + 3a2 = 0

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

Divide each side by '3'.
-0.3333333333 + -3.666666667a + a2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -3.666666667a + 0.3333333333 + a2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -3.666666667a + a2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -3.666666667a + a2 = 0 + 0.3333333333
-3.666666667a + a2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-3.666666667a + a2 = 0.3333333333

The a term is -3.666666667a.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667a + 3.361111114 + a2 = 0.3333333333 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667a + a2 = 0.3333333333 + 3.361111114

Combine like terms: 0.3333333333 + 3.361111114 = 3.6944444473
3.361111114 + -3.666666667a + a2 = 3.6944444473

Factor a perfect square on the left side:
(a + -1.833333334)(a + -1.833333334) = 3.6944444473

Calculate the square root of the right side: 1.922093767

Break this problem into two subproblems by setting 
(a + -1.833333334) equal to 1.922093767 and -1.922093767.

Subproblem 1

a + -1.833333334 = 1.922093767 Simplifying a + -1.833333334 = 1.922093767 Reorder the terms: -1.833333334 + a = 1.922093767 Solving -1.833333334 + a = 1.922093767 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + a = 1.922093767 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = 1.922093767 + 1.833333334 a = 1.922093767 + 1.833333334 Combine like terms: 1.922093767 + 1.833333334 = 3.755427101 a = 3.755427101 Simplifying a = 3.755427101

Subproblem 2

a + -1.833333334 = -1.922093767 Simplifying a + -1.833333334 = -1.922093767 Reorder the terms: -1.833333334 + a = -1.922093767 Solving -1.833333334 + a = -1.922093767 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + a = -1.922093767 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = -1.922093767 + 1.833333334 a = -1.922093767 + 1.833333334 Combine like terms: -1.922093767 + 1.833333334 = -0.088760433 a = -0.088760433 Simplifying a = -0.088760433

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.755427101, -0.088760433}

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