2a(3a-9)=2a-10(a-1)

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Solution for 2a(3a-9)=2a-10(a-1) equation: 


Simplifying
2a(3a + -9) = 2a + -10(a + -1)

Reorder the terms:
2a(-9 + 3a) = 2a + -10(a + -1)
(-9 * 2a + 3a * 2a) = 2a + -10(a + -1)
(-18a + 6a2) = 2a + -10(a + -1)

Reorder the terms:
-18a + 6a2 = 2a + -10(-1 + a)
-18a + 6a2 = 2a + (-1 * -10 + a * -10)
-18a + 6a2 = 2a + (10 + -10a)

Reorder the terms:
-18a + 6a2 = 10 + 2a + -10a

Combine like terms: 2a + -10a = -8a
-18a + 6a2 = 10 + -8a

Solving
-18a + 6a2 = 10 + -8a

Solving for variable 'a'.

Reorder the terms:
-10 + -18a + 8a + 6a2 = 10 + -8a + -10 + 8a

Combine like terms: -18a + 8a = -10a
-10 + -10a + 6a2 = 10 + -8a + -10 + 8a

Reorder the terms:
-10 + -10a + 6a2 = 10 + -10 + -8a + 8a

Combine like terms: 10 + -10 = 0
-10 + -10a + 6a2 = 0 + -8a + 8a
-10 + -10a + 6a2 = -8a + 8a

Combine like terms: -8a + 8a = 0
-10 + -10a + 6a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + -5a + 3a2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-5 + -5a + 3a2)' equal to zero and attempt to solve:

Simplifying
-5 + -5a + 3a2 = 0

Solving
-5 + -5a + 3a2 = 0

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

Divide each side by '3'.
-1.666666667 + -1.666666667a + a2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -1.666666667a + 1.666666667 + a2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -1.666666667a + a2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -1.666666667a + a2 = 0 + 1.666666667
-1.666666667a + a2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-1.666666667a + a2 = 1.666666667

The a term is -1.666666667a.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667a + 0.6944444447 + a2 = 1.666666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667a + a2 = 1.666666667 + 0.6944444447

Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117
0.6944444447 + -1.666666667a + a2 = 2.3611111117

Factor a perfect square on the left side:
(a + -0.8333333335)(a + -0.8333333335) = 2.3611111117

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(a + -0.8333333335) equal to 1.536590743 and -1.536590743.


Subproblem 1
a + -0.8333333335 = 1.536590743

Simplifying
a + -0.8333333335 = 1.536590743

Reorder the terms:
-0.8333333335 + a = 1.536590743

Solving
-0.8333333335 + a = 1.536590743

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + a = 1.536590743 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + a = 1.536590743 + 0.8333333335
a = 1.536590743 + 0.8333333335

Combine like terms: 1.536590743 + 0.8333333335 = 2.3699240765
a = 2.3699240765

Simplifying
a = 2.3699240765


Subproblem 2
a + -0.8333333335 = -1.536590743

Simplifying
a + -0.8333333335 = -1.536590743

Reorder the terms:
-0.8333333335 + a = -1.536590743

Solving
-0.8333333335 + a = -1.536590743

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + a = -1.536590743 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + a = -1.536590743 + 0.8333333335
a = -1.536590743 + 0.8333333335

Combine like terms: -1.536590743 + 0.8333333335 = -0.7032574095
a = -0.7032574095

Simplifying
a = -0.7032574095


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.3699240765, -0.7032574095}
Solution
a = {2.3699240765, -0.7032574095}

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