Math Problem Statement

Which of the following sets are equal? 2. 𝐴 = {1, 2, 3},𝐡 = {π‘₯; π‘₯^2 βˆ’ 2π‘₯ + 1 = 0}, 𝐢 = {π‘₯; π‘₯^3 βˆ’ 6π‘₯^2 +11π‘₯ βˆ’6 = 0}.

Solution

To determine which sets are equal, let's analyze the sets individually:

Set AA:

A={1,2,3}A = \{1, 2, 3\}

Set BB:

B={x;x2βˆ’2x+1=0}B = \{x; x^2 - 2x + 1 = 0\} Factorizing the quadratic equation: x2βˆ’2x+1=(xβˆ’1)2x^2 - 2x + 1 = (x - 1)^2 Thus, the solution is: x=1x = 1 So: B={1}B = \{1\}

Set CC:

C={x;x3βˆ’6x2+11xβˆ’6=0}C = \{x; x^3 - 6x^2 + 11x - 6 = 0\} Factorizing the cubic equation: x3βˆ’6x2+11xβˆ’6=(xβˆ’1)(xβˆ’2)(xβˆ’3)x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) The solutions are: x=1,2,3x = 1, 2, 3 Thus: C={1,2,3}C = \{1, 2, 3\}

Comparing the Sets:

  • A={1,2,3}A = \{1, 2, 3\}
  • B={1}B = \{1\}
  • C={1,2,3}C = \{1, 2, 3\}

Conclusion:

  • A=CA = C
  • Aβ‰ BA \neq B
  • Bβ‰ CB \neq C

Final Answer:

The sets AA and CC are equal.


Would you like detailed explanations of the factorization steps or have any related questions? Here are some related questions to deepen your understanding:

  1. How do you determine the roots of a cubic equation?
  2. Can you explain how to factorize higher-degree polynomials systematically?
  3. Why is x2βˆ’2x+1x^2 - 2x + 1 a perfect square trinomial?
  4. What are some real-world applications of solving polynomial equations?
  5. How can the equality of sets be determined in more complex scenarios?

Tip: Always verify your solutions by substituting the roots back into the original equation.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Algebra
Factorization of Polynomials

Formulas

Factoring a quadratic equation: x^2 - 2x + 1 = (x - 1)^2
Factoring a cubic equation: x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)

Theorems

Roots of Polynomial Equations
Equality of Sets

Suitable Grade Level

Grades 9-12