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When calculating the diameter equivalent ($\text{de}$) of a round duct in a HVAC system, and we know the rectangular duct size, we can use the following formula:

$\text{de} = 1.30 \times \frac{(a \times b)^{0.625}}{(a + b)^{0.25}}$

BUT IF we know $\text{de}$, and we know $a$, what formula could be used to solve for $b$? For testing purposes, I am assuming $\text{de}$ is 18" and $a$ is 16".

Using WolframAlpha...I was able to achieve an answer but they graphed to achieve the solution by using the intersecting lines...but I would like to solve this without graphing.

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    Tha$n$ks guys...I'll study LaTeX syntax before posting a question next time...2012-02-25

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Raise both sides to the 8th power. You get: $a^5 b^5 -(\frac{de}{1.3})^8 (a^2 + 2ab + b^2)=0$

Note that once you find the roots of the equation, you should check them to make sure they work. Since we raised both sides to the 8th power, we may have introduced sign errors.

In general, there is no formula for, or even a good way to write down, the roots of a quintic polynomial. You can plug the equation into your favorite software for a numerical approximation, but solutions to quintic equations can not always be expressed in terms of rational numbers, addition/subtraction, multiplication and radical symbols.

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    OK, solution has been updated to address the updated problem.2012-02-25