3a^2-2a-6=0

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Solution for 3a^2-2a-6=0 equation:


Simplifying
3a2 + -2a + -6 = 0

Reorder the terms:
-6 + -2a + 3a2 = 0

Solving
-6 + -2a + 3a2 = 0

Solving for variable 'a'.

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

Divide each side by '3'.
-2 + -0.6666666667a + a2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.6666666667a + 2 + a2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.6666666667a + a2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.6666666667a + a2 = 0 + 2
-0.6666666667a + a2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.6666666667a + a2 = 2

The a term is -0.6666666667a.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667a + 0.1111111112 + a2 = 2 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667a + a2 = 2 + 0.1111111112

Combine like terms: 2 + 0.1111111112 = 2.1111111112
0.1111111112 + -0.6666666667a + a2 = 2.1111111112

Factor a perfect square on the left side:
(a + -0.3333333334)(a + -0.3333333334) = 2.1111111112

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(a + -0.3333333334) equal to 1.452966315 and -1.452966315.

Subproblem 1

a + -0.3333333334 = 1.452966315 Simplifying a + -0.3333333334 = 1.452966315 Reorder the terms: -0.3333333334 + a = 1.452966315 Solving -0.3333333334 + a = 1.452966315 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = 1.452966315 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = 1.452966315 + 0.3333333334 a = 1.452966315 + 0.3333333334 Combine like terms: 1.452966315 + 0.3333333334 = 1.7862996484 a = 1.7862996484 Simplifying a = 1.7862996484

Subproblem 2

a + -0.3333333334 = -1.452966315 Simplifying a + -0.3333333334 = -1.452966315 Reorder the terms: -0.3333333334 + a = -1.452966315 Solving -0.3333333334 + a = -1.452966315 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = -1.452966315 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = -1.452966315 + 0.3333333334 a = -1.452966315 + 0.3333333334 Combine like terms: -1.452966315 + 0.3333333334 = -1.1196329816 a = -1.1196329816 Simplifying a = -1.1196329816

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.7862996484, -1.1196329816}

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