2*3a^2+3a+1=60

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Solution for 2*3a^2+3a+1=60 equation:


Simplifying
2 * 3a2 + 3a + 1 = 60

Multiply 2 * 3
6a2 + 3a + 1 = 60

Reorder the terms:
1 + 3a + 6a2 = 60

Solving
1 + 3a + 6a2 = 60

Solving for variable 'a'.

Reorder the terms:
1 + -60 + 3a + 6a2 = 60 + -60

Combine like terms: 1 + -60 = -59
-59 + 3a + 6a2 = 60 + -60

Combine like terms: 60 + -60 = 0
-59 + 3a + 6a2 = 0

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

Divide each side by '6'.
-9.833333333 + 0.5a + a2 = 0

Move the constant term to the right:

Add '9.833333333' to each side of the equation.
-9.833333333 + 0.5a + 9.833333333 + a2 = 0 + 9.833333333

Reorder the terms:
-9.833333333 + 9.833333333 + 0.5a + a2 = 0 + 9.833333333

Combine like terms: -9.833333333 + 9.833333333 = 0.000000000
0.000000000 + 0.5a + a2 = 0 + 9.833333333
0.5a + a2 = 0 + 9.833333333

Combine like terms: 0 + 9.833333333 = 9.833333333
0.5a + a2 = 9.833333333

The a term is 0.5a.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5a + 0.0625 + a2 = 9.833333333 + 0.0625

Reorder the terms:
0.0625 + 0.5a + a2 = 9.833333333 + 0.0625

Combine like terms: 9.833333333 + 0.0625 = 9.895833333
0.0625 + 0.5a + a2 = 9.895833333

Factor a perfect square on the left side:
(a + 0.25)(a + 0.25) = 9.895833333

Calculate the square root of the right side: 3.145764348

Break this problem into two subproblems by setting 
(a + 0.25) equal to 3.145764348 and -3.145764348.

Subproblem 1

a + 0.25 = 3.145764348 Simplifying a + 0.25 = 3.145764348 Reorder the terms: 0.25 + a = 3.145764348 Solving 0.25 + a = 3.145764348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 3.145764348 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 3.145764348 + -0.25 a = 3.145764348 + -0.25 Combine like terms: 3.145764348 + -0.25 = 2.895764348 a = 2.895764348 Simplifying a = 2.895764348

Subproblem 2

a + 0.25 = -3.145764348 Simplifying a + 0.25 = -3.145764348 Reorder the terms: 0.25 + a = -3.145764348 Solving 0.25 + a = -3.145764348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -3.145764348 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -3.145764348 + -0.25 a = -3.145764348 + -0.25 Combine like terms: -3.145764348 + -0.25 = -3.395764348 a = -3.395764348 Simplifying a = -3.395764348

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.895764348, -3.395764348}

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