10x^2+xy-4y^2=

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Solution for 10x^2+xy-4y^2= equation: 


Simplifying
10x2 + xy + -4y2 = 0

Reorder the terms:
xy + 10x2 + -4y2 = 0

Solving
xy + 10x2 + -4y2 = 0

Solving for variable 'x'.

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

Divide each side by '10'.
0.1xy + x2 + -0.4y2 = 0

Move the constant term to the right:

Add '0.4y2' to each side of the equation.
0.1xy + x2 + -0.4y2 + 0.4y2 = 0 + 0.4y2

Combine like terms: -0.4y2 + 0.4y2 = 0.0
0.1xy + x2 + 0.0 = 0 + 0.4y2
0.1xy + x2 = 0 + 0.4y2
Remove the zero:
0.1xy + x2 = 0.4y2

The x term is xy.  Take half its coefficient (0.5y).
Square it (0.25y2) and add it to both sides.

Add '0.25y2' to each side of the equation.
0.1xy + x2 + 0.25y2 = 0.4y2 + 0.25y2

Combine like terms: 0.4y2 + 0.25y2 = 0.65y2
0.1xy + x2 + 0.25y2 = 0.65y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 0.65y2

Calculate the square root of the right side: 0.806225775y

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


Subproblem 1
x + 0.5y = 0.806225775y

Simplifying
x + 0.5y = 0.806225775y

Solving
x + 0.5y = 0.806225775y

Solving for variable 'x'.

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

Add '-0.5y' to each side of the equation.
x + 0.5y + -0.5y = 0.806225775y + -0.5y

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

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

Simplifying
x = 0.306225775y


Subproblem 2
x + 0.5y = -0.806225775y

Simplifying
x + 0.5y = -0.806225775y

Solving
x + 0.5y = -0.806225775y

Solving for variable 'x'.

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

Add '-0.5y' to each side of the equation.
x + 0.5y + -0.5y = -0.806225775y + -0.5y

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

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

Simplifying
x = -1.306225775y


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

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