Multi-variable quadratic question

In summary, the conversation discusses finding a second height that satisfies a formula found in the first part of a problem. The solution involves multiplying and rearranging the formula to get a quadratic equation with h3 as the variable. The discriminant is a perfect square, making it easier to find the two solutions: h3 = h2-h1 and h3 = h1.
  • #1
Ragoza
3
0

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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  • #2
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)
 
  • #3
Wow, thank you. It's been awhile since I've done this stuff!
 
  • #4
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:

[tex] h_3^2 - h_2 h_3 + h_1 (h_2 -h_1) = 0 [/tex]

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:

[tex] h_2^2 - 4h_1(h_2 - h_1) [/tex]

[tex] = h_2^2 - 4h_1h_2 + 4h_1^2 [/tex]

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

1. What is a multi-variable quadratic equation?

A multi-variable quadratic equation is an algebraic equation that contains more than one variable and has a degree of two. It is a polynomial equation that can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.

2. How do you solve a multi-variable quadratic equation?

To solve a multi-variable quadratic equation, you can use various methods such as factoring, completing the square, or using the quadratic formula. The method used depends on the form of the equation and the availability of the variables. It is important to pay attention to the signs and simplify the equation before solving for the variables.

3. What are the applications of multi-variable quadratic equations?

Multi-variable quadratic equations have many real-life applications in fields such as physics, engineering, economics, and computer science. They can be used to model and solve problems related to motion, optimization, financial analysis, and data analysis.

4. How do you graph a multi-variable quadratic equation?

To graph a multi-variable quadratic equation, you can plot points by substituting different values for the variables and then connecting them with a smooth curve. The graph of a quadratic equation is a parabola, and the shape and position of the parabola can be determined by the coefficients of the equation.

5. What are some common mistakes when solving multi-variable quadratic equations?

Some common mistakes when solving multi-variable quadratic equations include forgetting to simplify the equation, making calculation errors, and misinterpreting the signs of the variables. It is important to check the solutions by substituting them back into the original equation to avoid common mistakes.

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