# solve for a in terms of b, c, and d

##### New member
I have the following equality:

a^2 / (a + b) = d^2 / (c + d), where a & d are unknown, and b & c are constants.

I would like to solve for a in terms of b, c, and d (isolate a in this equality).

This is part of a larger problem I'm working on, but I'm completely stuck right here and drawing a blank on how to handle this.

Any help would be appreciated. Thanks!

#### Subhotosh Khan

##### Super Moderator
Staff member
I have the following equality:

a^2 / (a + b) = d^2 / (c + d), where a & d are unknown, and b & c are constants.

I would like to solve for a in terms of b, c, and d (isolate a in this equality).

This is part of a larger problem I'm working on, but I'm completely stuck right here and drawing a blank on how to handle this.

Any help would be appreciated. Thanks!
a^2 / (a + b) = d^2 / (c + d)

a^2 * (c+d) = d^2 * (a + b)

Expand above and you'll have a quadratic equation in "a". Solve using your favorite method.

##### New member
Thanks, I did that originally but was thrown off with the a*d^2 and d*a^2 terms that resulted. Don't know why I didn't think to just complete the square. I get:

a = SQRT[ ( b*d / (c + d) + d^4 / (4* (c + d)^2 ) ] + d^2 / (2 * (c + d) )

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#### Subhotosh Khan

##### Super Moderator
Staff member
Thanks, I did that originally but was thrown off with the a*d^2 and d*a^2 terms that resulted. Don't know why I didn't think to just complete the square. I get:..........a = SQRT[ ( b*d / (c + d) + d^4 / (4* (c + d)^2 ) ] + d^2 / (2 * (c + d) )
Using quadratic formula, I get (which is derived from completing square):

a^2 / (a + b) = d^2 / (c + d)

a^2 * (c+d) = d^2 * (a + b)

a^2 * (c+d) - a*d^2 - b*d^2 = 0

a = $$\displaystyle \frac{d^2 \pm \sqrt{d^4 + 4* b *(c+d)*d^2}}{2 * (c+d)}$$

I did not check it - but probably those are equivalent.