Cross Multiplication in a Proportion [closed]

Solve for $$h$$:

$$\frac{h+z}{h} = \frac{a}{b}$$

I'm not sure how to simplify this with cross multiplication.

closed as off-topic by KReiser, José Carlos Santos, Namaste, Key Flex, Deepesh MeenaSep 28 '18 at 1:38

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• Is the left side $h + \frac{z}{h}$ or $\frac{h+z}{h}$? – Sean Roberson Sep 27 '18 at 21:33
• $\frac {h+z}{h} = \frac ab$ or $h + \frac {z}{h} = \frac ab$? Formatting is important. – Doug M Sep 27 '18 at 21:36
• (h+z)/h = a/b sorry about that – Katelyn Sep 27 '18 at 21:42
• Using MathJax could avoid confusion like this. – KM101 Sep 27 '18 at 22:29

$$\frac{h+z}{h} = \frac{a}{b}$$
$$\implies b(h+z) = ah$$
$$bh+bz = ah \implies bz = ah-bh \implies bz = (a-b)h \implies h = \frac{bz}{a-b}$$
Multiply both sides of the equation by $$h$$ and you get: $$\\h^2-\frac{a}{b}h+z=0$$ This should be easy to solve as it's just a quadratic.
$$\frac{h^2}{h}+\frac{z}{h} = \frac{a}{b}$$ $$\frac{h^2+z}{h} = \frac{a}{b}$$ $$h^2+z = \frac{a}{b}h$$ $$h^2-\frac{a}{b}h+z=0$$ Using the quadratic formula, we can get $$h = \frac{\frac{a}{b}\pm\sqrt{\left(\frac{a}{b}\right)^2-4z}}{2}$$