# Germany | Can you solve this ? | A Nice Math Olympiad Algebra Problem

TLDRThis educational video presents a complex Math Olympiad algebra problem: solving for x in the equation x² - x³ = 12. The presenter meticulously guides viewers through the problem-solving process, starting with rearranging the equation and applying algebraic identities to simplify it. They cleverly transform the equation into a form that allows the use of the perfect cube formula, leading to a quadratic equation. The video concludes with finding both real and complex solutions for x, validating the real solution by substitution. The presenter encourages viewers to practice such problems and subscribe for more educational content.

### Takeaways

- 🧮 The problem presented is an algebraic equation: \( x^2 - x^3 = 12 \).
- ➡️ The first step involves moving 12 to the left side to form the equation \( x^2 - x^3 - 12 = 0 \).
- 🔢 The equation is then split into terms with corresponding exponents: \( x^2 - 2^2 - x^3 + 2^3 = 0 \).
- 📚 The script uses the identity \( a^2 - b^2 = (a + b)(a - b) \) to factor the equation.
- 🔄 The equation is rearranged to \( (x^2 - 2^2) - (x^3 - 2^3) = 0 \) and factored further.
- 📐 The factored form is \( (x + 2)(x - 2)(x^2 + 2x + 4) = 0 \), indicating a perfect cube and a quadratic equation.
- 🔑 The solution for \( x + 2 = 0 \) gives the real solution \( x = -2 \).
- 🔍 The quadratic equation \( x^2 + 3x - 6 = 0 \) is solved using the quadratic formula.
- 🌐 The quadratic formula yields complex solutions: \( x = 3 \pm \frac{\sqrt{15}i}{2} \).
- 🔚 The final solutions are \( x = -2 \), \( x = 3 + \frac{\sqrt{15}i}{2} \), and \( x = 3 - \frac{\sqrt{15}i}{2} \).

### Q & A

### What is the original algebra problem presented in the video?

-The original algebra problem is to solve for x in the equation \( x^2 - x^3 = 12 \).

### How does the video begin the process of solving the equation?

-The video begins by moving the constant 12 to the left side of the equation to form \( x^2 - x^3 - 12 = 0 \).

### What mathematical concept is used to split the -12 into terms with corresponding exponents?

-The video uses the concept of factoring by grouping to split the -12 into terms with corresponding exponents, resulting in \( x^2 - 4x^3 - 8x^4 = 0 \).

### How does the video simplify the equation using the same power of two?

-The video simplifies the equation by combining like terms and factoring out common factors, resulting in \( (x^2 - 2^2) - (x^3 - 2^3) = 0 \).

### What algebraic identity is applied to further simplify the equation?

-The video applies the algebraic identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) to simplify the equation.

### What is the form of the equation after applying the perfect cube rule?

-After applying the perfect cube rule, the equation is in the form \( (x + 2)(x - 2)(x^2 + 2x + 4) = 0 \).

### What are the three values of x found by solving the equation?

-The values of x found are \( x = -2 \) (real solution) and \( x = \frac{-3 \pm \sqrt{15}i}{2} \) (complex solutions).

### How does the video verify the real solution \( x = -2 \)?

-The video verifies the real solution by substituting \( x = -2 \) back into the original equation and confirming that both sides equal 12.

### What is the significance of the complex solutions found in the problem?

-The complex solutions indicate that the equation has roots that are not real numbers, which is a common occurrence in polynomial equations of degree three or higher.

### How does the video conclude the problem-solving process?

-The video concludes by confirming that all the solutions (real and complex) satisfy the original equation, thus completing the problem-solving process.

### Outlines

### 🧮 Solving the Cubic Equation x² - x³ = 12

This paragraph introduces a math problem involving a cubic equation, x² - x³ = 12. The solution process begins by moving the constant 12 to the left side, resulting in the equation x² - x³ - 12 = 0. The goal is to find the values of x that satisfy this equation. The approach taken involves breaking down the equation into terms with corresponding exponents, simplifying, and then applying algebraic identities to further simplify the equation into a form that can be solved for x.

### 🔍 Applying Algebraic Identities to Solve the Equation

The paragraph continues the solution process by applying algebraic identities to simplify the equation. The equation is first factored into a form that resembles a perfect cube, x² - 2², and then further into a form that resembles a perfect cube with a power of three, (x + 2)(x² - 2x + 4) = 0. The solution then involves finding the roots of the quadratic equation x² - 2x + 4 = 0, which yields a real solution x = -2 and complex solutions involving the square root of -15. The paragraph concludes with a verification step, where the real solution is substituted back into the original equation to confirm its correctness.

### 📝 Conclusion and Verification of the Solutions

The final paragraph wraps up the solution process by verifying the solutions obtained. The real solution x = -2 is checked by substituting it back into the original equation, confirming that it satisfies the equation. The paragraph also acknowledges the complex solutions obtained and thanks the viewers for watching. It encourages viewers to subscribe to the channel for more content, signaling the end of the video script.

### Mindmap

### Keywords

### 💡Math Olympiad

### 💡Algebra

### 💡Exponents

### 💡Equation

### 💡Perfect Cube

### 💡Factoring

### 💡Quadratic Equation

### 💡Complex Solutions

### 💡Quadratic Formula

### 💡Imaginary Unit

### Highlights

Introduction to solving a Math Olympiad Algebra problem involving x² - x³ = 12.

Rearranging the equation to x² - x³ = -12 and then isolating the variable terms.

Splitting the constant -12 into powers of 2 to match the variable terms.

Grouping like terms and forming a new equation x² - 2² - x³ - 2³ = 0.

Factoring out common terms to simplify the equation.

Applying the perfect cube formula to further simplify the equation.

Identifying the equation as a difference of cubes and applying the formula.

Solving for x by setting up a quadratic equation.

Finding the real solution x = -2 by setting one factor equal to zero.

Using the quadratic formula to solve for the remaining complex solutions.

Calculating the discriminant (b² - 4ac) for the quadratic equation.

Applying the square root to the discriminant to find complex solutions.

Determining the complex solutions involving the square root of a negative number.

Listing all the solutions: real solution x = -2 and complex solutions involving imaginary numbers.

Verifying the real solution by substituting it back into the original equation.

Confirming the correctness of the real solution through substitution.

Encouraging viewers to subscribe for more educational content.

Closing the video with a reminder to subscribe and a farewell.

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