MHB How to Understand the Ratio of Quadratic Roots?

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Understanding the ratio of quadratic roots involves solving for coefficients, particularly when dealing with monic quadratics. Participants emphasize the importance of these skills for Olympiad preparation, suggesting that brute force methods can be a fallback if time permits. The discussion highlights the necessity of showing work to clarify misunderstandings about the ratios and solutions. Overall, the conversation stresses both strategic approaches and the need for clarity in problem-solving. Mastery of these concepts is crucial for success in competitive mathematics.
Mathsonfire
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Mathsonfire said:
86

Those are nice problems. If you're prepping for the Olympiad, you will need the skills. If all else fails, brute force it! Of course, if you're not fast enough, you'll waste too much time.

Personally, I wrote down the ratios and solved for b. You could solve for any of the 4 remaining arbitrary coefficients.

They were nice to you and made them both monic quadratics.
 
tkhunny said:
Those are nice problems. If you're prepping for the Olympiad, you will need the skills. If all else fails, brute force it! Of course, if you're not fast enough, you'll waste too much time.

Personally, I wrote down the ratios and solved for b. You could solve for any of the 4 remaining arbitrary coefficients.

They were nice to you and made them both monic quadratics.

I didnt get it
 
Mathsonfire said:
I didnt get it

Didn't get what? Ratios? Solve for...? Please provide YOUR work.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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