solving x^4+1=0 | Summary and Q&A

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January 17, 2021
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blackpenredpen
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solving x^4+1=0

TL;DR

Learn how to factor a polynomial equation without using complex numbers.

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Questions & Answers

Q: How can you factor the equation x^4 + 1 = 0 without using complex numbers?

The first method involves recognizing a sum of two squares and applying the difference of two squares formula. By adding 2 times x^2 and 1, and then subtracting it, we can factor the equation as (x^2 - √2x + 1)(x^2 + √2x + 1).

Q: What are the solutions to the equation x^4 + 1 = 0?

The solutions are x = (√2 + √2i)/2, x = (-√2 + √2i)/2, x = (√2 - √2i)/2, and x = (-√2 - √2i)/2.

Q: How does the second method of factoring use complex numbers?

The second method involves treating the equation as a difference of two squares and using the fact that i^2 = -1. By factoring x^4 - i^2 = 0, we get (x^2 - i)(x^2 + i) = 0.

Q: How can the square root of i be determined?

The square root of i can be found by treating it as a nested square root and considering the real part of i (0) and the imaginary part (√-1). By applying the proper manipulation, the square root of i is found to be ±(√2/2 + √2i/2).

Summary & Key Takeaways

  • The video demonstrates two methods to factor the equation x^4 + 1 = 0 without resorting to complex numbers.

  • The first method involves recognizing a sum of two squares and applying the difference of two squares formula.

  • The second method utilizes complex numbers and a nested square root to find the factors of the equation.

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