Multi-variable quadratic question

AI Thread Summary
The discussion revolves around solving a multi-variable quadratic equation related to heights in a physics problem. The equation h1(h2-h1)=h3(h2-h3) is manipulated to express h3 in terms of h1 and h2. By rearranging and expanding the equation, a quadratic form is derived: h3^2 - h2h3 + h1(h2 - h1) = 0. The solutions include h3 = h2 - h1 and h3 = h1, with the discriminant revealing a perfect square that simplifies the solution process. The conversation highlights the importance of proper variable manipulation in quadratic equations.
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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!
 
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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)
 
Wow, thank you. It's been awhile since I've done this stuff!
 
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.
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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