3a(2a-5)=130-2(a+12)

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Solution for 3a(2a-5)=130-2(a+12) equation:


Simplifying
3a(2a + -5) = 130 + -2(a + 12)

Reorder the terms:
3a(-5 + 2a) = 130 + -2(a + 12)
(-5 * 3a + 2a * 3a) = 130 + -2(a + 12)
(-15a + 6a2) = 130 + -2(a + 12)

Reorder the terms:
-15a + 6a2 = 130 + -2(12 + a)
-15a + 6a2 = 130 + (12 * -2 + a * -2)
-15a + 6a2 = 130 + (-24 + -2a)

Combine like terms: 130 + -24 = 106
-15a + 6a2 = 106 + -2a

Solving
-15a + 6a2 = 106 + -2a

Solving for variable 'a'.

Reorder the terms:
-106 + -15a + 2a + 6a2 = 106 + -2a + -106 + 2a

Combine like terms: -15a + 2a = -13a
-106 + -13a + 6a2 = 106 + -2a + -106 + 2a

Reorder the terms:
-106 + -13a + 6a2 = 106 + -106 + -2a + 2a

Combine like terms: 106 + -106 = 0
-106 + -13a + 6a2 = 0 + -2a + 2a
-106 + -13a + 6a2 = -2a + 2a

Combine like terms: -2a + 2a = 0
-106 + -13a + 6a2 = 0

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

Divide each side by '6'.
-17.66666667 + -2.166666667a + a2 = 0

Move the constant term to the right:

Add '17.66666667' to each side of the equation.
-17.66666667 + -2.166666667a + 17.66666667 + a2 = 0 + 17.66666667

Reorder the terms:
-17.66666667 + 17.66666667 + -2.166666667a + a2 = 0 + 17.66666667

Combine like terms: -17.66666667 + 17.66666667 = 0.00000000
0.00000000 + -2.166666667a + a2 = 0 + 17.66666667
-2.166666667a + a2 = 0 + 17.66666667

Combine like terms: 0 + 17.66666667 = 17.66666667
-2.166666667a + a2 = 17.66666667

The a term is -2.166666667a.  Take half its coefficient (-1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
-2.166666667a + 1.173611113 + a2 = 17.66666667 + 1.173611113

Reorder the terms:
1.173611113 + -2.166666667a + a2 = 17.66666667 + 1.173611113

Combine like terms: 17.66666667 + 1.173611113 = 18.840277783
1.173611113 + -2.166666667a + a2 = 18.840277783

Factor a perfect square on the left side:
(a + -1.083333334)(a + -1.083333334) = 18.840277783

Calculate the square root of the right side: 4.340538882

Break this problem into two subproblems by setting 
(a + -1.083333334) equal to 4.340538882 and -4.340538882.

Subproblem 1

a + -1.083333334 = 4.340538882 Simplifying a + -1.083333334 = 4.340538882 Reorder the terms: -1.083333334 + a = 4.340538882 Solving -1.083333334 + a = 4.340538882 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + a = 4.340538882 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + a = 4.340538882 + 1.083333334 a = 4.340538882 + 1.083333334 Combine like terms: 4.340538882 + 1.083333334 = 5.423872216 a = 5.423872216 Simplifying a = 5.423872216

Subproblem 2

a + -1.083333334 = -4.340538882 Simplifying a + -1.083333334 = -4.340538882 Reorder the terms: -1.083333334 + a = -4.340538882 Solving -1.083333334 + a = -4.340538882 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + a = -4.340538882 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + a = -4.340538882 + 1.083333334 a = -4.340538882 + 1.083333334 Combine like terms: -4.340538882 + 1.083333334 = -3.257205548 a = -3.257205548 Simplifying a = -3.257205548

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.423872216, -3.257205548}

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