Multi-variable quadratic question

[Ragoza]

Homework Statement

At the end of a much longer problem, I'm asked to find a second height that will satisfy a formula found for a height in the first part of the problem where:

Homework Equations

h₁(h₂-h₁)=h₃(h₂-h₃)

The Attempt at a Solution

I know the answer I should get: h₃=h₂-h₁

But I cannot figure how to manipulate the variables to get that. I've tried using the quadratic equation but get lost under the square root sign. This has got to be easier than I'm making it!

 

[LowlyPion]

Welcome to PF.

Multiply it out both sides and rearrange:

h2h1 -h1² = h2h3 - h3²

h2h1 - h2h3 = h2(h1 - h3) = h1² - h3² = (h1 + h3)(h1 - h3)

divide by (h1-h3)

 

[Ragoza]

Wow, thank you. It's been awhile since I've done this stuff!

 

[cepheid]

You want to get a quadratic with h3 as the variable and the other heights as constants (because you want h3 *in terms of* those other two heights). So expand the right hand side (but not the left) and rearrange:

$$h_3^2 - h_2 h_3 + h_1 (h_2 -h_1) = 0$$

There's your quadratic. a = 1, b = h2, c = left hand side of the original equation.

Now, it's messy, but the two solutions you'll get are the one you're expecting, and another one, namely h3 = h1 (which is obviously a solution, by inspection).

Hint: Your discriminant is:

$$h_2^2 - 4h_1(h_2 - h_1)$$

$$= h_2^2 - 4h_1h_2 + 4h_1^2$$

This is a *perfect square*, making things really easy.