3a^2=6a-1

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


Simplifying
3a2 = 6a + -1

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

Solving
3a2 = -1 + 6a

Solving for variable 'a'.

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

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

Combine like terms: -1 + 1 = 0
1 + -6a + 3a2 = 0 + 6a + -6a
1 + -6a + 3a2 = 6a + -6a

Combine like terms: 6a + -6a = 0
1 + -6a + 3a2 = 0

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

Divide each side by '3'.
0.3333333333 + -2a + a2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2a + -0.3333333333 + a2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2a + a2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2a + a2 = 0 + -0.3333333333
-2a + a2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2a + a2 = -0.3333333333

The a term is -2a.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2a + 1 + a2 = -0.3333333333 + 1

Reorder the terms:
1 + -2a + a2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + -2a + a2 = 0.6666666667

Factor a perfect square on the left side:
(a + -1)(a + -1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(a + -1) equal to 0.816496581 and -0.816496581.

Subproblem 1

a + -1 = 0.816496581 Simplifying a + -1 = 0.816496581 Reorder the terms: -1 + a = 0.816496581 Solving -1 + a = 0.816496581 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = 0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + a = 0.816496581 + 1 a = 0.816496581 + 1 Combine like terms: 0.816496581 + 1 = 1.816496581 a = 1.816496581 Simplifying a = 1.816496581

Subproblem 2

a + -1 = -0.816496581 Simplifying a + -1 = -0.816496581 Reorder the terms: -1 + a = -0.816496581 Solving -1 + a = -0.816496581 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = -0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + a = -0.816496581 + 1 a = -0.816496581 + 1 Combine like terms: -0.816496581 + 1 = 0.183503419 a = 0.183503419 Simplifying a = 0.183503419

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.816496581, 0.183503419}

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