Notes on Olympiad Challenge: Solving the Equation a^4 = (a - 1)^4

Introduction

• Presenter: Mathematics enthusiast
• Topic: Common mistakes in solving an Olympiad challenge involving exponents

The Challenge

• Equation to solve: a^4 = (a - 1)^4
• Common student mistake: Taking the fourth root directly leads to errors.

Step-by-Step Explanation

Step 1: Rewrite the Equation

• Rewrite the equation: a^4 - (a - 1)^4 = 0

Step 2: Preparing for Difference of Squares

• Represent each side using squares:
  • (a^2)^2 = ((a - 1)^2)^2
• Thus, we can form a difference of squares:
  • a^2 - (a - 1)^2 imes (a^2 + (a - 1)^2) = 0

Step 3: Applying the Difference of Squares Formula

• Formula: x² - y² = (x - y)(x + y)
• First parentheses after applying the formula:
  • (a^2 - (a - 1)^2)
• Second parentheses remains:
  • (a^2 + (a - 1)^2)

Step 4: Simplifying the First Parenthesis

• Expanding (a - 1)²:
  • a² - 2a + 1
• Thus, simplifying gives:
  • 2a - 1

Step 5: Simplifying the Second Parenthesis

• The expression will lead to:
  • 2a² - 2a + 1 = 0

Step 6: Finding Roots Using the Quadratic Formula

• Discriminant calculation for 2a² - 2a + 1 = 0:
  • Coefficients: a = 2, b = -2, c = 1
  • D = b² - 4ac = (-2)² - 4(2)(1)
  • Results in a negative discriminant, hence complex roots.

Step 7: Roots Found

• Applying quadratic formula:
  • a = (-b ± √D) / 2a
  • Results in complex roots:
    • a = 1 ± i/2
• First root found from the first set is a = 1/2.

Conclusion

• Final answers:
  • Root 1: a = 1/2
  • Root 2: a = 1 + i/2
  • Root 3: a = 1 - i/2
• Common mistakes highlighted: Students often overlook complex roots by taking fourth roots incorrectly without recognizing all roots.

Closing Thoughts

• Understanding complex numbers and roots is crucial in solving higher-level math challenges.
• Encouragement to continue learning math and tackling challenging problems.
• Reminder to like and subscribe if the content was helpful.