A tricky problem with a "divine" answer!
TLDRIn this episode of 'Mind Your Decisions', Preshtalwalkar tackles a complex mathematical problem involving real values of x. By introducing clever substitutions, setting the first term as 'a' and the second as 'b', the equation simplifies to 'a + b = x'. Through algebraic manipulations, including a difference of squares and factoring, the solution converges to the golden ratio, a divine answer that highlights the beauty of mathematics. The video concludes by verifying the solution's validity, emphasizing the importance of thorough checking in problem-solving.
Takeaways
- 🔢 The problem involves solving an equation with square root expressions.
- 📐 The equation is transformed using substitutions: a for \(\sqrt{x-1/x}\) and b for \(\sqrt{1-1/x}\).
- 🔄 By multiplying the equation \(a + b = x\) by \(a - b\), a difference of squares is formed.
- 📘 Simplifying the left side results in \(a^2 - b^2 = x(a - b)\).
- ✅ The equation is further simplified to \(x - 1/x = a - b\).
- 🔄 Flipping and simplifying the equation leads to \(2a = x + 1 - 1/x\).
- 🔢 The quadratic equation \(2a = a^2 + 1\) is derived and solved.
- 🏵️ The solution \(a = 1\) is found, which implies \(\sqrt{x-1/x} = 1\).
- 🔍 Squaring both sides of \(\sqrt{x-1/x} = 1\) results in a quadratic equation \(x^2 - 1 = x\).
- 🌟 The golden ratio \((1 + \sqrt{5})/2\) and its negative reciprocal are found as potential solutions.
- ❌ The negative reciprocal is discarded as it does not satisfy the original equation.
- 🏅 The golden ratio is confirmed as the solution to the original equation.
Q & A
What is the given equation in the problem?
-The equation is (x - 1/x)^(1/2) + (1 - 1/x)^(1/2) = x.
What substitutions are made to simplify the equation?
-The first term (x - 1/x)^(1/2) is substituted as 'a' and the second term (1 - 1/x)^(1/2) is substituted as 'b'. The equation then becomes a + b = x.
What mathematical operation is applied after the substitution?
-Both sides of the equation are multiplied by (a - b), simplifying the left-hand side to a difference of squares: a^2 - b^2 = x * (a - b).
How is the difference of squares simplified?
-It is simplified as a^2 = x - 1/x and b^2 = 1 - 1/x, so a^2 - b^2 = x - 1.
What is the next step after finding the expression for a^2 - b^2?
-The expression is substituted into the equation, and both sides are divided by x, leading to (x - 1)/x = a - b.
How is the equation involving a and b further simplified?
-The equation (x - 1)/x = a - b is added to a + b = x, which cancels the b terms, resulting in 2a = x + 1 - 1/x.
What quadratic equation is derived from the expression for 'a'?
-From 2a = a^2 + 1, the quadratic equation a^2 - 2a + 1 = 0 is derived.
What are the solutions to the quadratic equation for 'a'?
-The quadratic equation has a double root, a = 1.
How is 'x' solved from the equation involving 'a'?
-Since a = (x - 1/x)^(1/2), squaring both sides gives x^2 - 1 = x, which simplifies to x^2 - x - 1 = 0.
What are the solutions for 'x' in the final equation?
-The solutions are the golden ratio (1 + sqrt(5))/2 and the negative reciprocal of the golden ratio (1 - sqrt(5))/2.
Which solution for 'x' is valid, and why?
-The golden ratio is the valid solution because the sum of two square roots must be non-negative, while the negative reciprocal results in a negative value for x, which doesn't satisfy the original equation.
Outlines
🔢 Solving the Golden Ratio Equation
The video script introduces a mathematical problem involving the real values of x in an equation involving square roots and fractions. The host, Preshtalwalkar, suggests a clever substitution method to solve the equation. The terms are assigned to variables 'a' and 'b', simplifying the equation to a plus b equals x. By multiplying both sides by 'a - b' and simplifying, the equation transforms into a difference of squares. The solution process involves recognizing the pattern \(a^2 - b^2 = x - 1\), which leads to a quadratic equation in terms of 'a'. Solving this quadratic equation yields a double root, indicating 'a' equals one. Substituting back, the equation simplifies to a quadratic form in 'x', which is solved using the quadratic formula, yielding the golden ratio and its negative reciprocal as potential solutions. After evaluating these solutions in the original equation, the golden ratio is confirmed as the correct solution, concluding the problem-solving process.
Mindmap
Keywords
💡Substitution
💡Difference of squares
💡Quadratic equation
💡Golden ratio
💡Reciprocal
💡Square root
💡Real values of x
💡Quadratic formula
💡Factoring
💡Double root
Highlights
Introduction to solving a tricky mathematical problem involving real values of x.
Equation presented: (x - 1)/√x + (1 - 1/x)/√x = x.
Suggestion to pause for attempting the problem independently.
Use of clever substitutions to simplify the equation.
Introduction of variables a and b for substitution.
Transformation of the equation into a plus b equals x.
Multiplication of both sides by (a - b) to simplify.
Application of the difference of squares formula.
Simplification to a^2 - b^2 = x(a - b).
Derivation of a^2 and b^2 in terms of x.
Substitution of a^2 - b^2 with x - 1.
Division of both sides by x to further simplify.
Rearrangement and simplification to 2a = x + 1 - 1/x.
Identification of a^2 in the equation.
Solving the quadratic equation 2a = a^2 + 1.
Finding the double root a = 1.
Relating a back to the original equation's terms.
Squaring both sides to solve for x.
Deriving the quadratic equation x^2 - x - 1 = 0.
Application of the quadratic formula to find roots.
Identification of the golden ratio as a potential solution.
Elimination of the negative root as a non-viable solution.
Confirmation of the golden ratio as the solution to the original equation.
Conclusion emphasizing the 'divine' nature of the solution.