(5a-3a^2)-(-6a-6)=0

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Solution for (5a-3a^2)-(-6a-6)=0 equation:


Simplifying
(5a + -3a2) + -1(-6a + -6) = 0

Remove parenthesis around (5a + -3a2)
5a + -3a2 + -1(-6a + -6) = 0

Reorder the terms:
5a + -3a2 + -1(-6 + -6a) = 0
5a + -3a2 + (-6 * -1 + -6a * -1) = 0
5a + -3a2 + (6 + 6a) = 0

Reorder the terms:
6 + 5a + 6a + -3a2 = 0

Combine like terms: 5a + 6a = 11a
6 + 11a + -3a2 = 0

Solving
6 + 11a + -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 + -3.666666667a + a2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -3.666666667a + 2 + a2 = 0 + 2

Reorder the terms:
-2 + 2 + -3.666666667a + a2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -3.666666667a + a2 = 0 + 2
-3.666666667a + a2 = 0 + 2

Combine like terms: 0 + 2 = 2
-3.666666667a + a2 = 2

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 = 2 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667a + a2 = 2 + 3.361111114

Combine like terms: 2 + 3.361111114 = 5.361111114
3.361111114 + -3.666666667a + a2 = 5.361111114

Factor a perfect square on the left side:
(a + -1.833333334)(a + -1.833333334) = 5.361111114

Calculate the square root of the right side: 2.315407332

Break this problem into two subproblems by setting 
(a + -1.833333334) equal to 2.315407332 and -2.315407332.

Subproblem 1

a + -1.833333334 = 2.315407332 Simplifying a + -1.833333334 = 2.315407332 Reorder the terms: -1.833333334 + a = 2.315407332 Solving -1.833333334 + a = 2.315407332 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 = 2.315407332 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = 2.315407332 + 1.833333334 a = 2.315407332 + 1.833333334 Combine like terms: 2.315407332 + 1.833333334 = 4.148740666 a = 4.148740666 Simplifying a = 4.148740666

Subproblem 2

a + -1.833333334 = -2.315407332 Simplifying a + -1.833333334 = -2.315407332 Reorder the terms: -1.833333334 + a = -2.315407332 Solving -1.833333334 + a = -2.315407332 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 = -2.315407332 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = -2.315407332 + 1.833333334 a = -2.315407332 + 1.833333334 Combine like terms: -2.315407332 + 1.833333334 = -0.482073998 a = -0.482073998 Simplifying a = -0.482073998

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.148740666, -0.482073998}

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