Prompt for Solving Mathematical Equations

Please give an example of the incorrect results.

Posted by: @ssadvisor

↑

Please give an example of the incorrect results.

Let's solve the equation step by step.

Given equation:
\[
4x^2 + 12x + \frac{12}{x} + \frac{4}{x^2} = 47
\]

### Step 1: Multiply by \(x^2\) to eliminate the fractions
To simplify the equation, multiply every term by \(x^2\) (assuming \(x \neq 0\)):
\[
4x^4 + 12x^3 + 12x + 4 = 47x^2
\]

### Step 2: Rearrange the equation
Bring all terms to one side to form a polynomial equation:
\[
4x^4 + 12x^3 - 47x^2 + 12x + 4 = 0
\]

### Step 3: Factor or use the Rational Root Theorem
To find the roots, we can attempt to factor the polynomial or apply the Rational Root Theorem. The Rational Root Theorem suggests possible rational roots, which are the factors of the constant term (±1, ±2, ±4) divided by the factors of the leading coefficient (±1, ±2, ±4).

Testing these possible rational roots, we find that \(x = 1\) is a root.

### Step 4: Perform polynomial division
To factor the polynomial, divide the original polynomial by \(x - 1\):

Performing the division:

\[
\text{Divide } 4x^4 + 12x^3 - 47x^2 + 12x + 4 \text{ by } (x - 1)
\]
Gives:
\[
4x^3 + 16x^2 - 31x - 4
\]

### Step 5: Continue factoring
We now factor the cubic polynomial \(4x^3 + 16x^2 - 31x - 4\).

Using synthetic division or the Rational Root Theorem again, we can test for more roots. After further analysis, we find that \(x = \frac{1}{2}\) is another root.

Divide the cubic polynomial by \(x - \frac{1}{2}\), and we obtain:
\[
4x^2 + 18x - 8
\]

### Step 6: Solve the quadratic equation
Finally, solve the quadratic equation:
\[
4x^2 + 18x - 8 = 0
\]
Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substitute \(a = 4\), \(b = 18\), and \(c = -8\):
\[
x = \frac{-18 \pm \sqrt{18^2 - 4(4)(-8)}}{2(4)}
\]
\[
x = \frac{-18 \pm \sqrt{324 + 128}}{8}
\]
\[
x = \frac{-18 \pm \sqrt{452}}{8}
\]
\[
x = \frac{-18 \pm 2\sqrt{113}}{8}
\]
Simplify:
\[
x = \frac{-9 \pm \sqrt{113}}{4}
\]

### Step 7: Compile all roots
The roots of the original equation are:
1. \(x = 1\)
2. \(x = \frac{1}{2}\)
3. \(x = \frac{-9 + \sqrt{113}}{4}\)
4. \(x = \frac{-9 - \sqrt{113}}{4}\)

### Conclusion:
The equation \(4x^2 + 12x + \frac{12}{x} + \frac{4}{x^2} = 47\) has four roots:
- One rational root: \(x = 1\)
- One rational root: \(x = \frac{1}{2}\)
- Two irrational roots: \(x = \frac{-9 + \sqrt{113}}{4}\) and \(x = \frac{-9 - \sqrt{113}}{4}\)

This solution accounts for all possible roots, ensuring every step is correct and every potential solution is covered.

@mathros You don't want the output to be in this format, or is the answer incorrect?

Posted by: @husein

↑

You don't want the output to be in this format, or is the answer incorrect?

answer incorrect

@mathros If you want the LLM to have a high success rate, you'll have to create specific prompts for specific topics in math, not a generic calculator. As much as you improve the prompt if it's generic, it'll always answer incorrectly in some math fields.

I tried to do something just for linear algebra; however, some questions gave me wrong answers. So i tried creating prompts for each chapter I'm working on and this helped improve the output and generate correct responses.

@husein Bro, give an example of prompts for someone chapter please

@mathros I couldn't find any of the prompts right now, but as a general idea in every prompt, I explained the chapter thoroughly and what I'll be asking about. Then, before asking each question, I give a sample question answer where the answer was correct. 

This wasn't the most efficient way to do it, but that's why I did it at that time. I tried it on 15 questions, and it only gave me one wrong answer.

@husein Got it, friend. I will also follow your idea, maybe something will work out. If you achieve any positive result, please let me know

@husein It was possible to slightly improve the prompt, but it does not give the desired result.

You are a leading expert in solving mathematical equations of any complexity. I have an extremely important task for you: solve the following [equation] with absolute accuracy. Your solution should consider and present all possible roots, including all rational, irrational, complex, and repeated roots, leaving no solution overlooked. Provide a detailed, step-by-step solution, ensuring that every intermediate step is clearly explained and thoroughly verified. Continue calculations until a complete, finalized solution is reached, leaving no part of the equation unresolved. Avoid mentioning any numerical methods or software; focus solely on providing a fully accurate, mathematically justified result. The final result must be 100% correct, covering all aspects of the equation, regardless of its type or complexity. Here is a list of types of equations you can solve:

1. **Linear Equations**
Solve an equation of the form \( ax + b = 0 \), where \( a \) and \( b \) are coefficients. Indicate all possible solutions for the unknown variable \( x \).

2. **Quadratic Equations**
Solve an equation of the form \( ax^2 + bx + c = 0 \), using standard formulas for the roots, factoring, or discriminant analysis, considering all possible roots, including complex ones.

3. **Higher-Order Equations**
Find the solution for an equation of the form \( ax^n + bx^{n-1} + \dots + k = 0 \), carefully analyzing all possible roots, including repeated and complex roots.

4. **Rational Equations**
Solve equations with fractions where algebraic expressions may appear in the numerator and denominator. Optimize the equation by finding all possible solutions after simplification.

5. **Irrational Equations**
Find all possible solutions for an equation that includes roots, providing a full analysis of the roots obtained after raising to the appropriate power.

6. **Exponential Equations**
Solve an equation of the form \( a^x = b \), indicating all possible values of the variable \( x \) after logarithmic transformation.

7. **Logarithmic Equations**
Find all possible solutions for an equation that contains logarithms by converting it to the corresponding exponential form and thoroughly checking the result.

8. **Trigonometric Equations**
Solve an equation containing trigonometric functions, considering all possible periodic solutions and their general form.

9. **Diophantine Equations**
Find all possible integer solutions for an equation by applying factorization and analyzing the properties of numbers.

10. **Differential Equations**
Find all possible general or particular solutions of a differential equation, providing a complete analysis of the results.

Provide the best possible solution for this [equation] and present it in LaTeX format.

[equation]="4\cdot x^2+12\cdot x+\frac{12}{x}+\frac{4}{x^2}=47"

Prompt engineering experts where are you? Has no one encountered similar problems?

@mathros Hi Friend, there are Limitations in LLMs when it comes to reasoning and Math. 

In many cases, it is not the prompt; it is the LLM itself. It will not generate the correct answers. Remember, it is AI, and it may hallucinate.

you will need to test the same prompt with multiple LLMs, try sonnet, gemini, and gpt-4 , and see if you get different results.

try Qwen2-vl 2b it works for me for calculus, coding etc. response is great so far. You can use it as a private llm