How to solve for complicated term + squared term (partial correlation formula)

[Maziana]

The partial correlation formula:

p = (A - BC)/((sqrt(1 - B²))(sqrt(1 - C²)))

where p = partial correlation (r_23.1), A = r₂₃, B = r₁₂, C = r₁₃

I want to solve for B.

I tried solving it by squaring the whole formula first:
p² = (A - BC) (A - BC) / (1 - B²) (1 - C²)
p² = (A² - 2ABC + B²C²) / (1 - C² - B² + B²C²)

Then multiply by p²:
p² - p²C² - p²B² + p²B²C² = A² - 2ABC + B²C²

Then I tried moving every term with an B to one side:
- p²B² + p²B²C² + 2ABC - B²C²= - p² + p²C² + A²

But then I am stumped. I don't know what to do next, since there are 3 terms with B² and one with B on the same. Any help would be greatly appreciated.

Not sure if this belongs in the algebra category (maybe this involves calculus?).

 

[Ishuda]

The partial correlation formula:

p = (A - BC)/((sqrt(1 - B²))(sqrt(1 - C²)))

where p = partial correlation (r_23.1), A = r₂₃, B = r₁₂, C = r₁₃

I want to solve for B.

I tried solving it by squaring the whole formula first:
p² = (A - BC) (A - BC) / (1 - B²) (1 - C²)
p² = (A² - 2ABC + B²C²) / (1 - C² - B² + B²C²)

Then multiply by p²:
p² - p²C² - p²B² + p²B²C² = A² - 2ABC + B²C²

Then I tried moving every term with an B to one side:
- p²B² + p²B²C² + 2ABC - B²C²= - p² + p²C² + A²

But then I am stumped. I don't know what to do next, since there are 3 terms with B² and one with B on the same. Any help would be greatly appreciated.

Not sure if this belongs in the algebra category (maybe this involves calculus?).

What you would like to do is to get an equation of the form
a B² + b B + c = 0
where a, b, c may be, in themselves, complicated expressions but don't depend on B. You can then use the quadratic formula. So, collect terms and you have
(p²C² - p² - C²) B² + (2AC) B + (p² - p²C² - A²) = 0

 

[Maziana]

Yay, it works! Thanks so much, both of you.

((-2*A*C) + (sqrt( (2*A*C)^2 - (4*((p^2*C^2) - p^2 - C^2)*(p^2 - (p^2*C^2) - A^2))
)))/ (2*((p^2*C^2) - p^2 - C^2))

 

[abeha]

But then I am stumped. I don't know what to do next, since there are 3 terms with B² and one with B on the same. Any help would be greatly appreciated.