Maths Olympiad Question | A nice algebraic equation | You should know this trick!

Summary

The video script presents a mathematical problem involving the evaluation of an expression with x raised to the fourth power. The presenter uses substitution by setting U = 1 - x and applies the binomial theorem to simplify and solve for x. After finding the value of U, they determine x has four possible solutions, showcasing the process with clear explanations and engaging the audience to follow along.

Takeaways

• 📚 The video discusses solving the equation \(x^4 + (2 - x)^4 = 34\) to find the value of x.
• ✏️ The approach begins by substituting \(U = 1 - x\), leading to transformations for both x and \(2 - x\).
• 🔄 Substituting these transformations into the original equation results in a binomial expansion using the binomial theorem.
• 🧮 The binomial expansion is detailed, using coefficients from the binomial theorem to expand \( (1 - U)^4 \) and \( (1 + U)^4 \).
• ➗ After expanding, terms are combined and simplified, leading to an equation in terms of U, which is then solved using factorization.
• 🧩 The factorized equation yields two quadratic equations, \( U^2 + 8 = 0 \) and \( U^2 - 2 = 0 \), each solved separately.
• ⚙️ The solutions for U are complex numbers for the first equation and real numbers for the second.
• 🔍 Substituting the values of U back into the expression \(x = 1 + U\), gives four possible solutions for x.
• 🧑‍🏫 The final solutions for x are \(x_1 = 1 + 2\sqrt{2}i\), \(x_2 = 1 - 2\sqrt{2}i\), \(x_3 = 1 + \sqrt{2}\), and \(x_4 = 1 - \sqrt{2}\).
• 👍 The video concludes by summarizing the four solutions, reminding viewers to follow and subscribe for more content.

Q & A

What is the first step taken to evaluate the equation x^4 + 2 - x^4 = 34?

The first step is to substitute u = 1 - x, allowing the equation to be simplified by replacing expressions involving x with u.

How is the value of 2 - x derived after substituting u = 1 - x?

After substituting u = 1 - x, the value of 2 - x is simplified to 1 + u.

Why is the binomial theorem used in this problem?

The binomial theorem is used to expand the expressions (1 - u)^4 and (1 + u)^4, which are crucial for simplifying and solving the equation.

What is the significance of the expansion of (1 - u)^4 and (1 + u)^4?

The expansion helps to identify and simplify terms, leading to a final equation that can be solved for u, and subsequently for x.

How are the terms of the binomial expansion combined to simplify the equation?

The like terms from the expansions of (1 - u)^4 and (1 + u)^4 are added together, which allows for the elimination of certain terms and further simplification of the equation.

What does the equation simplify to after factoring and canceling out common terms?

After factoring and canceling out common terms, the equation simplifies to u^4 + 6u^2 - 16 = 0.

How is the quadratic equation u^4 + 6u^2 - 16 = 0 solved?

The equation is solved by grouping and factorization, leading to two possible quadratic equations u^2 + 8 = 0 and u^2 - 2 = 0.

What are the possible values of u derived from the quadratic equations?

The possible values of u are u = ±2√2i from u^2 + 8 = 0 and u = ±√2 from u^2 - 2 = 0.

How are the values of x determined from the values of u?

The values of x are determined by substituting the values of u back into the equation x = 1 - u, resulting in four possible solutions for x.

What are the final four solutions for x?

The final four solutions for x are x1 = 1 + 2√2i, x2 = 1 - 2√2i, x3 = 1 + √2, and x4 = 1 - √2.