Simple Problem STUMPS PhotoMath! Can You Figure It Out?
TLDRIn this video, Presh Talwalkar tackles an algebra problem that challenges solvers like PhotoMath and Mathway. The problem involves finding all real numbers x for which (x^2 - 7x + 11)^(x^2 - 13x + 42) equals 1. Talwalkar explains that while most solvers find four solutions, there are actually six. He demonstrates how to solve the problem by considering cases where the base or exponent equals one or the base is negative one and the exponent is even, leading to the solutions x = 2, 3, 4, 5, 6, and 7.
Takeaways
- 🧠 The problem involves solving for x in the equation (x^2 - 7x + 11)^(x^2 - 13x + 42) = 1.
- 📱 Popular math-solving apps like Photomath, Mathway, and Symbolab struggle to find all solutions, missing some.
- 🔍 Wolfram Alpha is noted for successfully identifying all six solutions to the problem.
- 🤔 The presenter suggests solving the problem by considering different cases where the expression can equal one.
- 📝 The first case involves setting the base x^2 - 7x + 11 equal to 1, leading to solutions x = 2 and x = 5.
- 🎯 The second case considers the exponent x^2 - 13x + 42 equal to 0, yielding solutions x = 6 and x = 7.
- ⚠️ It's important to verify that the base is not zero to avoid an indeterminate form.
- 🔄 The third case examines when the base is -1, requiring the exponent to be even, resulting in solutions x = 3 and x = 4.
- 📉 The presenter uses graphing tools like Desmos to illustrate the solutions, although they initially suggest fewer solutions than actually exist.
- 😄 A humorous note is made about the product of all solutions, prompting viewers to read a specific sentence aloud.
- 🌟 The video concludes with a call to action for viewers to subscribe and support the channel for more free math content.
Q & A
What is the main challenge presented in the video?
-The main challenge is to solve for all real numbers x for which the equation \((x^2 - 7x + 11)^{(x^2 - 13x + 42)} = 1\) holds true.
Why is this algebra problem considered difficult?
-This problem is considered difficult because it stumps many computer solvers, including PhotoMath, which cannot provide step-by-step solutions and incorrectly suggests there are only four solutions when there are actually six.
What is the average time given to high school students in Massachusetts to solve similar problems?
-High school students in Massachusetts are given an average of 30 minutes to solve each question.
How many solutions does Wolfram Alpha find for the given problem?
-Wolfram Alpha is able to find all six solutions to the problem.
What are the different cases that can result in the equation equaling one?
-The different cases include: 1) The base equation equals one, 2) The exponent equals zero with a non-zero base, and 3) The base equals negative one and the exponent is an even number.
What are the two solutions obtained from setting the base equation equal to one?
-The solutions obtained by setting \(x^2 - 7x + 11 = 1\) are \(x = 2\) and \(x = 5\).
What are the two solutions obtained from setting the exponent equal to zero?
-The solutions obtained by setting \(x^2 - 13x + 42 = 0\) are \(x = 6\) and \(x = 7\).
Why is it necessary to check that the base is not equal to zero?
-It is necessary to check that the base is not equal to zero because \(0^0\) is an indeterminate form, which would not satisfy the original equation.
What are the two additional solutions found by setting the base equal to negative one?
-The solutions obtained by setting \(x^2 - 7x + 11 = -1\) and ensuring the exponent is even are \(x = 3\) and \(x = 4\).
What is the total number of solutions to the problem presented in the video?
-There are six solutions to the problem: \(x = 2, 3, 4, 5, 6,\) and \(7\).
What joke does the presenter leave the viewers with regarding the solutions to the problem?
-The joke is about multiplying all of the solutions together, which is a playful nod to the complexity of the problem.
Outlines
🧮 Algebra Problem Introduction
Presh Talwalkar introduces an algebra problem to solve for all real numbers x that satisfy the equation \(x^2 - 7x + 11\) raised to the power of \(x^2 - 13x + 42\) equals 1. This problem was given to high school students in Massachusetts with an average of 30 minutes per question. The video challenges viewers to solve it and provides a solution later in the video. The problem's complexity is highlighted by the fact that many computer solvers, including popular apps like Photomath, Mathway, and Symbolab, fail to find all six solutions, with some only identifying four. Only Wolfram Alpha successfully finds all six solutions.
Mindmap
Keywords
💡Algebra
💡PhotoMath
💡Exponent Rules
💡Factoring
💡Indeterminate Form
💡Wolfram Alpha
💡Desmos
💡Mind Your Decisions
💡Presh Talwalkar
💡Mathematical Puzzle
Highlights
Solve for all real numbers x in the equation x^2 - 7x + 11 * (x^2 - 13x + 42)^x = 1.
The problem was given to high school students in Massachusetts with an average of 30 minutes per question.
Many computer solvers, including Photomath, do not provide correct solutions for this problem.
Photomath, with over a hundred million downloads, cannot solve this problem and suggests there are four solutions.
Mathway and Symbolab also miss two of the six solutions.
Desmos graph suggests four solutions, but there are actually six.
Wolfram Alpha is able to find all six solutions.
To solve, set the base equation x^2 - 7x + 11 equal to 1 and factor for solutions.
The solutions from the base equation are x = 2 and x = 5.
Set the exponent equal to zero for another case, leading to x^2 - 13x + 42 = 0.
Solutions from the exponent equation are x = 6 and x = 7.
Ensure the base is not zero to avoid an indeterminate form.
Set the base equal to negative one and check if the exponent is even for the third case.
Solutions from the base equal to negative one are x = 3 and x = 4.
The six solutions to the problem are x = 2, 3, 4, 5, 6, and 7.
Multiplying all the solutions together leads to a humorous outcome.
The video is part of a series on YouTube that has over 100 million views and aims to inspire confidence in math.
The presenter encourages subscriptions and engagement with the content.