Solving Quadratic Equations with Binary Operator $\otimes$

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Discussion Overview

The discussion revolves around the definition and application of a binary operator $\otimes$ defined as $a\otimes b = a^2+b+9$. Participants explore the computation of specific examples using this operator, addressing potential arithmetic errors and suggesting methods for verification.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Post 1 presents calculations for $6\otimes 4$, $3\otimes 3$, $4\otimes 6$, and $i\otimes p$, asserting that the computations are correct but acknowledging that arithmetic errors can occur.
  • Post 2 identifies arithmetic errors in the calculations for $3\otimes 3$ and $4\otimes 6$, suggesting that using a programming function could help verify the results.
  • Post 5 provides corrected calculations for $3\otimes 3$ and $4\otimes 6$, indicating a revised result for $4\otimes 6$ as 31 instead of 21.
  • Post 6 reiterates the calculation for $3\otimes 3$, confirming the result as 21 but presenting a different addition step ($9+12$) which may imply a misunderstanding or error in the earlier arithmetic.
  • Post 4 discusses the environments in which Python can be used for calculations, emphasizing flexibility in tool choice.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correctness of the arithmetic in the calculations, particularly for $3\otimes 3$ and $4\otimes 6$. There is no consensus on the final correctness of all computations, as multiple viewpoints on the errors exist.

Contextual Notes

Some calculations appear to depend on the accuracy of arithmetic operations, and there are unresolved discrepancies in the results presented by different participants.

karush
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Define the binary operater $\otimes$ by
$a\otimes b = a^2+b+9$
Find
a. $6\otimes 4 = 6^2+4+9 = 36+13 = 49$
b. $3\otimes 3 = 3^2+3+9 = 9+13 = 22$
c $4\otimes 6 = 4^2+6+9 = 16+15 =21$
d. $i\otimes p = i^2+p+9$

i think its ok but most math errors are in simple arithmetic
 
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b and c have arithmetic errors.

The best way to check the answers is to write a function in some programming language and run it on your inputs. For example, in Python the function would be
Python:
def f(x, y):
  return x**2 + y + 9
Then you can run f(6, 4) and so on in an interpreter.
 
karush said:
Define the binary operater $\otimes$ by
$a\otimes b = a^2+b+9$
Find
a. $6\otimes 4 = 6^2+4+9 = 36+13 = 49$
b. $3\otimes 3 = 3^2+3+9 = 9+13 = 22$
c $4\otimes 6 = 4^2+6+9 = 16+15 =21$
d. $i\otimes p = i^2+p+9$

i think its ok but most math errors are in simple arithmetic
Evgeny.Makarov said:
b and c have arithmetic errors.

The best way to check the answers is to write a function in some programming language and run it on your inputs. For example, in Python the function would be
Python:
def f(x, y):
  return x**2 + y + 9
Then you can run f(6, 4) and so on in an interpreter.
Evgeny.Makarov said:
b and c have arithmetic errors.

The best way to check the answers is to write a function in some programming language and run it on your inputs. For example, in Python the function would be
Python:
def f(x, y):
  return x**2 + y + 9
Then you can run f(6, 4) and so on in an interpreter.
ummm is that under colcalc
at least I saw it there when one class was teaching us the programs
 
Python is a programming language that can be used in different environments: CoCalc, Jupyter Notebook, PyCharm by JetBrains or simply from the command line. Use whatever tool you are familiar with, or use any other programming language.

It's a good idea to quote only relevant passages.
 
Find (corrected)
a. $6\otimes 4 = 6^2+4+9 = 36+13 = 49$
b. $3\otimes 3 = 3^2+3+9 = 9+13 = 21$
c $4\otimes 6 = 4^2+6+9 = 16+15 =31$
d. $i\otimes p = i^2+p+9$
 
$3^2+3+9=9+12$.
 
gotcha
3\otimes 3 = 3^2+3+9 = 9+12= 21
 

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