a^2+ab-55b^2=

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Solution for a^2+ab-55b^2= equation:


Simplifying
a2 + ab + -55b2 = 0

Reorder the terms:
ab + a2 + -55b2 = 0

Solving
ab + a2 + -55b2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '55b2' to each side of the equation.
ab + a2 + -55b2 + 55b2 = 0 + 55b2

Combine like terms: -55b2 + 55b2 = 0
ab + a2 + 0 = 0 + 55b2
ab + a2 = 0 + 55b2
Remove the zero:
ab + a2 = 55b2

The a term is ab.  Take half its coefficient (0.5b).
Square it (0.25b2) and add it to both sides.

Add '0.25b2' to each side of the equation.
ab + a2 + 0.25b2 = 55b2 + 0.25b2

Combine like terms: 55b2 + 0.25b2 = 55.25b2
ab + a2 + 0.25b2 = 55.25b2

Factor a perfect square on the left side:
(a + 0.5b)(a + 0.5b) = 55.25b2

Calculate the square root of the right side: 7.433034374b

Break this problem into two subproblems by setting 
(a + 0.5b) equal to 7.433034374b and -7.433034374b.


Subproblem 1
a + 0.5b = 7.433034374b

Simplifying
a + 0.5b = 7.433034374b

Solving
a + 0.5b = 7.433034374b

Solving for variable 'a'.

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

Add '-0.5b' to each side of the equation.
a + 0.5b + -0.5b = 7.433034374b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = 7.433034374b + -0.5b
a = 7.433034374b + -0.5b

Combine like terms: 7.433034374b + -0.5b = 6.933034374b
a = 6.933034374b

Simplifying
a = 6.933034374b


Subproblem 2
a + 0.5b = -7.433034374b

Simplifying
a + 0.5b = -7.433034374b

Solving
a + 0.5b = -7.433034374b

Solving for variable 'a'.

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

Add '-0.5b' to each side of the equation.
a + 0.5b + -0.5b = -7.433034374b + -0.5b

Combine like terms: 0.5b + -0.5b = 0.0
a + 0.0 = -7.433034374b + -0.5b
a = -7.433034374b + -0.5b

Combine like terms: -7.433034374b + -0.5b = -7.933034374b
a = -7.933034374b

Simplifying
a = -7.933034374b


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {6.933034374b, -7.933034374b}

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