(6x+3at)(x+at)=0

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Solution for (6x+3at)(x+at)=0 equation: 


Simplifying
(6x + 3at)(x + at) = 0

Reorder the terms:
(3at + 6x)(x + at) = 0

Reorder the terms:
(3at + 6x)(at + x) = 0

Multiply (3at + 6x) * (at + x)
(3at * (at + x) + 6x * (at + x)) = 0
((at * 3at + x * 3at) + 6x * (at + x)) = 0

Reorder the terms:
((3atx + 3a2t2) + 6x * (at + x)) = 0
((3atx + 3a2t2) + 6x * (at + x)) = 0
(3atx + 3a2t2 + (at * 6x + x * 6x)) = 0
(3atx + 3a2t2 + (6atx + 6x2)) = 0

Reorder the terms:
(3atx + 6atx + 3a2t2 + 6x2) = 0

Combine like terms: 3atx + 6atx = 9atx
(9atx + 3a2t2 + 6x2) = 0

Solving
9atx + 3a2t2 + 6x2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '3'.
3(3atx + a2t2 + 2x2) = 0

Factor a trinomial.
3((at + x)(at + 2x)) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(at + x)' equal to zero and attempt to solve:

Simplifying
at + x = 0

Solving
at + x = 0

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

Add '-1x' to each side of the equation.
at + x + -1x = 0 + -1x

Combine like terms: x + -1x = 0
at + 0 = 0 + -1x
at = 0 + -1x
Remove the zero:
at = -1x

Divide each side by 't'.
a = -1t-1x

Simplifying
a = -1t-1x

Subproblem 2
Set the factor '(at + 2x)' equal to zero and attempt to solve:

Simplifying
at + 2x = 0

Solving
at + 2x = 0

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

Add '-2x' to each side of the equation.
at + 2x + -2x = 0 + -2x

Combine like terms: 2x + -2x = 0
at + 0 = 0 + -2x
at = 0 + -2x
Remove the zero:
at = -2x

Divide each side by 't'.
a = -2t-1x

Simplifying
a = -2t-1x
Solution
a = {-1t-1x, -2t-1x}

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