2x^2+ax-5a^2=0

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Solution for 2x^2+ax-5a^2=0 equation: 


Simplifying
2x2 + ax + -5a2 = 0

Reorder the terms:
ax + -5a2 + 2x2 = 0

Solving
ax + -5a2 + 2x2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-0.2ax + a2 + -0.4x2 = 0

Move the constant term to the right:

Add '0.4x2' to each side of the equation.
-0.2ax + a2 + -0.4x2 + 0.4x2 = 0 + 0.4x2

Combine like terms: -0.4x2 + 0.4x2 = 0.0
-0.2ax + a2 + 0.0 = 0 + 0.4x2
-0.2ax + a2 = 0 + 0.4x2
Remove the zero:
-0.2ax + a2 = 0.4x2

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.
-0.2ax + a2 + 0.25x2 = 0.4x2 + 0.25x2

Combine like terms: 0.4x2 + 0.25x2 = 0.65x2
-0.2ax + a2 + 0.25x2 = 0.65x2

Factor a perfect square on the left side:
(a + 0.5x)(a + 0.5x) = 0.65x2

Calculate the square root of the right side: 0.806225775x

Break this problem into two subproblems by setting 
(a + 0.5x) equal to 0.806225775x and -0.806225775x.


Subproblem 1
a + 0.5x = 0.806225775x

Simplifying
a + 0.5x = 0.806225775x

Solving
a + 0.5x = 0.806225775x

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 = 0.806225775x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
a + 0.0 = 0.806225775x + -0.5x
a = 0.806225775x + -0.5x

Combine like terms: 0.806225775x + -0.5x = 0.306225775x
a = 0.306225775x

Simplifying
a = 0.306225775x


Subproblem 2
a + 0.5x = -0.806225775x

Simplifying
a + 0.5x = -0.806225775x

Solving
a + 0.5x = -0.806225775x

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 = -0.806225775x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
a + 0.0 = -0.806225775x + -0.5x
a = -0.806225775x + -0.5x

Combine like terms: -0.806225775x + -0.5x = -1.306225775x
a = -1.306225775x

Simplifying
a = -1.306225775x


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.306225775x, -1.306225775x}

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