Math Problem Statement

Expand and simplify the expression (3a − 2b)(6a − b) + 4a(9a + 4b).

Solution

Let's expand and simplify the expression:

(3a2b)(6ab)+4a(9a+4b)(3a - 2b)(6a - b) + 4a(9a + 4b)

Step 1: Expand each part of the expression.

  1. Expand (3a2b)(6ab)(3a - 2b)(6a - b) using the distributive property: (3a)(6a)+(3a)(b)+(2b)(6a)+(2b)(b)(3a)(6a) + (3a)(-b) + (-2b)(6a) + (-2b)(-b) =18a23ab12ab+2b2= 18a^2 - 3ab - 12ab + 2b^2 Combine the like terms: 18a215ab+2b218a^2 - 15ab + 2b^2

  2. Expand 4a(9a+4b)4a(9a + 4b): 4a(9a)+4a(4b)=36a2+16ab4a(9a) + 4a(4b) = 36a^2 + 16ab

Step 2: Combine the expanded parts.

(18a215ab+2b2)+(36a2+16ab)(18a^2 - 15ab + 2b^2) + (36a^2 + 16ab)

Combine like terms:

  • For a2a^2 terms: 18a2+36a2=54a218a^2 + 36a^2 = 54a^2

  • For abab terms: -15ab + 16ab = 1ab \quad \text{(or simply ab)}

  • For b2b^2 terms: 2b22b^2

Final Simplified Expression:

54a2+ab+2b254a^2 + ab + 2b^2

This is the expanded and simplified form of the given expression.

Would you like more details or have any questions about this process?


Here are 5 related questions you might find interesting:

  1. How would the expansion change if the sign of bb were reversed in the original expression?
  2. What would happen if the coefficients of aa and bb were doubled in the original expression?
  3. How do you factor the expression 54a2+ab+2b254a^2 + ab + 2b^2?
  4. What is the geometric interpretation of the simplified expression?
  5. How would the solution change if the powers of aa and bb were different in the original expression?

Tip: When expanding and simplifying expressions, it helps to carefully track each term to avoid mistakes in combining like terms.

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Math Problem Analysis

Mathematical Concepts

Algebra
Distributive Property
Combining Like Terms

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12