Master the Pythagorean Theorem with Mind-Blowing Math Tricks! | IMO Trigonometry Problem

In this video, we tackle a challenging Olympiad math problem involving the Pythagorean Theorem and exponential equations. Watch as we solve the problem step-by-step on paper, providing a clear and detailed explanation without any voiceover. This unique approach allows you to focus entirely on the math and understand the process visually.
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✅ Key Concepts Covered
• Pythagorean Theorem
• Exponential Equations
• Quadratic Equations
• Algebraic Manipulations
• Problem-Solving Strategies
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✅ Detailed Steps
We begin with the equation based on the Pythagorean Theorem: (x^3)^2 = x^2 + (x^2)^2
1. Express in a simplified form: x^6 = x^2 + x^4
2. Divide both sides by x^2: x^4 = x^2 + 1
3. Rearrange to form a standard polynomial equation: x^4 - x^2 - 1 = 0
4. Let t = x^2, then the equation becomes: t^2 - t - 1 = 0
5. Solve the quadratic equation using the quadratic formula t = (-b ± √(b^2 - 4ac)) / 2a: t = (1 + √(5)) / 2 or t = (1 - √(5)) / 2
6. Since t = x^2, we consider: x^2 = (1 + √(5)) / 2
7. Take the square root of both sides to find x: x = √((1 + √(5)) / 2)
8. Evaluate the second root t = (1 - √(5)) / 2: Since (1 - √(5)) / 2 is negative, it is not acceptable for x^2.
Therefore, the solution is: x = √((1 + √(5)) / 2)
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✅ Learning Outcomes
By the end of this video, you will:
• Understand how to apply the Pythagorean Theorem to complex problems.
• Learn to manipulate and solve exponential equations.
• Gain skills in solving quadratic equations using the quadratic formula.
• Enhance your problem-solving strategies for advanced math competitions.
• Develop a deeper understanding of algebraic manipulations and their applications.
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✅ Why Subscribe to SciMarvels?
At SciMarvels, we are dedicated to making complex math and physics problems accessible and enjoyable. Our unique approach of solving problems without voiceovers allows you to concentrate solely on the math, fostering a deeper understanding. Subscribe to join a community of learners who are passionate about mastering scientific concepts through clear, step-by-step solutions.
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✅ Tags
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[fatemetahernezhad]

The concept of the Pythagorean theorem is fascinating to understand.

[yosefsalehi-bhzy]

That was nice...at first i though both roots will be imaginary but the outcome was different