3a^2+a=6

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Solution for 3a^2+a=6 equation:


Simplifying
3a2 + a = 6

Reorder the terms:
a + 3a2 = 6

Solving
a + 3a2 = 6

Solving for variable 'a'.

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

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

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

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

Move the constant term to the right:

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

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

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

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

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333a + 0.25 + a2 = 2 + 0.25

Reorder the terms:
0.25 + 0.3333333333a + a2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + 0.3333333333a + a2 = 2.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(a + 0.5) equal to 1.5 and -1.5.

Subproblem 1

a + 0.5 = 1.5 Simplifying a + 0.5 = 1.5 Reorder the terms: 0.5 + a = 1.5 Solving 0.5 + a = 1.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.5 + -0.5 a = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 a = 1 Simplifying a = 1

Subproblem 2

a + 0.5 = -1.5 Simplifying a + 0.5 = -1.5 Reorder the terms: 0.5 + a = -1.5 Solving 0.5 + a = -1.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.5 + -0.5 a = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 a = -2 Simplifying a = -2

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1, -2}

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