3x^2+ax-a^2=0

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Solution for 3x^2+ax-a^2=0 equation: 


Simplifying
3x2 + ax + -1a2 = 0

Reorder the terms:
ax + -1a2 + 3x2 = 0

Solving
ax + -1a2 + 3x2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1ax + a2 + -3x2 = 0

Move the constant term to the right:

Add '3x2' to each side of the equation.
-1ax + a2 + -3x2 + 3x2 = 0 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-1ax + a2 + 0 = 0 + 3x2
-1ax + a2 = 0 + 3x2
Remove the zero:
-1ax + a2 = 3x2

The a term is ax.  Take half its coefficient (0.5x).
Square it (0.25x2) and add it to both sides.

Add '0.25x2' to each side of the equation.
-1ax + a2 + 0.25x2 = 3x2 + 0.25x2

Combine like terms: 3x2 + 0.25x2 = 3.25x2
-1ax + a2 + 0.25x2 = 3.25x2

Factor a perfect square on the left side:
(a + 0.5x)(a + 0.5x) = 3.25x2

Calculate the square root of the right side: 1.802775638x

Break this problem into two subproblems by setting 
(a + 0.5x) equal to 1.802775638x and -1.802775638x.


Subproblem 1
a + 0.5x = 1.802775638x

Simplifying
a + 0.5x = 1.802775638x

Solving
a + 0.5x = 1.802775638x

Solving for variable 'a'.

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

Add '-0.5x' to each side of the equation.
a + 0.5x + -0.5x = 1.802775638x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
a + 0.0 = 1.802775638x + -0.5x
a = 1.802775638x + -0.5x

Combine like terms: 1.802775638x + -0.5x = 1.302775638x
a = 1.302775638x

Simplifying
a = 1.302775638x


Subproblem 2
a + 0.5x = -1.802775638x

Simplifying
a + 0.5x = -1.802775638x

Solving
a + 0.5x = -1.802775638x

Solving for variable 'a'.

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

Add '-0.5x' to each side of the equation.
a + 0.5x + -0.5x = -1.802775638x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
a + 0.0 = -1.802775638x + -0.5x
a = -1.802775638x + -0.5x

Combine like terms: -1.802775638x + -0.5x = -2.302775638x
a = -2.302775638x

Simplifying
a = -2.302775638x


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {1.302775638x, -2.302775638x}

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