# Viral Video | A Trending Math Olympiad Problem | Can You Solve This Math Problem In Your Head?

TLDRIn this tutorial, the presenter tackles a challenging math problem involving the equation x^4 = 16. Initially, they attempt an unconventional approach but switch to a standard method to find the roots. They manipulate the equation into a form that allows the application of the difference of squares formula, leading to a quadratic equation. The roots are found using the quadratic formula, resulting in four solutions: x1 = โ2 * (1 - i), x2 = โ2 * (1 + i), x3 = โ2 * (-1 - i), and x4 = โ2 * (-1 + i). The presenter also hints at an alternative method yielding simpler roots and invites viewers to request a video on that method.

### Takeaways

- ๐งฉ The video presents a math challenge involving solving the equation x^4 = 16.
- ๐ The host initially tried a different approach but decided to revert to a standard method for solving exponential equations.
- ๐ The process involves transforming the equation into a form that allows for factoring and the application of the difference of squares.
- ๐ The equation is manipulated to x^4 + 16 = 0 and then rearranged to set up the difference of squares.
- ๐ The difference of squares formula a^2 - b^2 = (a+b)(a-b) is applied to the equation.
- ๐ The quadratic equation derived from the process is solved using the quadratic formula.
- ๐ The roots of the equation are found to be complex numbers involving i, the imaginary unit.
- ๐ The video mentions a second method that yields simpler roots, which the host is willing to demonstrate if requested.
- ๐ The final solution includes four roots, two of which are positive and two negative, all involving the square root of 2 and i.
- ๐ The script encourages viewers to subscribe to the channel and engage by commenting if they have alternative solutions or wish to see the second method.
- ๐ The host expresses appreciation for the viewers and invites them to be part of the channel's community.

### Q & A

### What is the main math problem presented in the video?

-The main math problem presented in the video is to find the value of x that satisfies the equation x^4 = 16.

### What was the initial approach the presenter considered for solving the problem?

-The initial approach the presenter considered involved moving the 16 to the other side of the equation and changing it to a negative sign, but this approach was discarded as it only yielded two roots.

### Why was the initial approach discarded by the presenter?

-The initial approach was discarded because it was not as comprehensive as the presenter wanted, providing only two roots instead of the expected four for the equation.

### What method did the presenter eventually decide to use to solve the equation?

-The presenter decided to use a more straightforward method involving algebraic manipulation and the difference of squares to solve the equation.

### How many roots did the presenter find for the equation using the chosen method?

-The presenter found four roots for the equation using the chosen method.

### What is the significance of the 'difference of squares' in solving the equation?

-The 'difference of squares' is significant as it simplifies the equation into a form that can be solved using the quadratic formula, leading to the discovery of all four roots.

### What is the quadratic formula mentioned in the video?

-The quadratic formula is x = (-b ยฑ โ(bยฒ - 4ac)) / (2a), which is used to solve quadratic equations of the form axยฒ + bx + c = 0.

### How does the presenter use the quadratic formula to find the roots of the equation?

-The presenter applies the quadratic formula by identifying the coefficients a, b, and c from the equation, and then substituting them into the formula to find the roots.

### What are the four roots the presenter found for the equation x^4 = 16?

-The four roots found by the presenter are โ2 * (1 - i), โ2 * (1 + i), โ2 * (-1 - i), and โ2 * (-1 + i).

### Does the presenter mention an alternative method to solve the equation?

-Yes, the presenter mentions an alternative method that yields roots of the form ยฑ2i, but does not elaborate on it in the video.

### How can viewers request the presenter to demonstrate the alternative method?

-Viewers can request the presenter to demonstrate the alternative method by commenting 'second method' in the comment section of the video.

### Outlines

### ๐ Introduction to the Math Challenge

The video begins with a warm welcome and an introduction to a mathematical challenge involving the equation \( x^4 = -16 \). The host mentions a previous similar challenge and decides to use a standard approach to solve the exponential equation. The audience is encouraged to subscribe for more content, and the host promises to address an alternative method if requested in the comments.

### ๐ Exploring the Equation and Attempting a Solution

The host starts by rearranging the equation to \( x^4 + 16 = 0 \) and attempts to solve it using a method involving the introduction of \( 8x^2 \) to form a perfect square trinomial. However, the initial approach leads to confusion, and the host decides to revert to a more straightforward method involving the difference of squares. The explanation includes algebraic manipulation aimed at simplifying the equation into a form that can be solved using basic quadratic formulas.

### ๐งฉ Applying the Difference of Squares and Solving the Quadratic

The host applies the difference of squares formula to the equation, resulting in a quadratic form that can be solved using the quadratic formula. The process involves identifying coefficients \( a \), \( b \), and \( c \) and then substituting them into the formula to find the roots. The host simplifies the equation step by step, eventually factoring out common terms and isolating the variable \( x \) to find two of the roots.

### ๐ Concluding the Solution and Inviting Further Engagement

The host concludes by finding all four roots of the original equation, which are presented in a simplified form. The video ends with an invitation for viewers to suggest alternative methods or to request a demonstration of the second method mentioned earlier. The host expresses appreciation for the viewers and signs off with a reminder of the channel's mission and love for the audience.

### Mindmap

### Keywords

### ๐กMath Olympiad Problem

### ๐กExponential Equation

### ๐กQuadratic Equation

### ๐กRoots

### ๐กSquare Root

### ๐กImaginary Unit (i)

### ๐กDifference of Squares

### ๐กComplex Numbers

### ๐กQuadratic Formula

### ๐กFactorization

### Highlights

Introduction to solving the math problem x^4 = 16 using a different approach than previously used.

Explanation of the initial attempt to solve the problem and the decision to discard that approach.

The standard approach to solving exponential equations is chosen for clarity and effectiveness.

Transforming the equation x^4 = 16 into a solvable form by moving terms around.

Introduction of a new element to the equation to facilitate solving.

Grouping terms in the equation to simplify and prepare for further manipulation.

Application of the difference of squares formula to the equation.

Rewriting the equation to express it in terms of x^2 and constants.

Simplification of the equation to a quadratic form for easier solving.

Use of the quadratic formula to find potential solutions for x.

Derivation of the roots of the equation using algebraic manipulation.

Identification of the first two roots of the equation involving square roots and iota.

Introduction of the second case to find the remaining roots of the equation.

Application of the quadratic formula again to find the third and fourth roots.

Final expression of all four roots that satisfy the original exponential challenge.

Mention of an alternative method that yields simpler roots and an invitation for viewers to request a video on this method.

Closing remarks encouraging viewers to engage with the content and the channel.