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Difficulty: HardSimplifying Algebraic Expressions

Consider the algebraic expression:

−2a(3a−4b)−3b(a−2b)+5a2-2a(3a - 4b) - 3b(a - 2b) + 5a^2

Which of the following represents the correctly simplified form of this expression?

  1. −a2+5ab+6b2-a^2 + 5ab + 6b^2Answer
  2. B
    −a2+5ab−6b2-a^2 + 5ab - 6b^2
  3. C
    −a2−11ab+6b2-a^2 - 11ab + 6b^2
  4. D
    4ab+6b24ab + 6b^2

Answer

The correctly simplified form is −a2+5ab+6b2-a^2 + 5ab + 6b^2.
The expression correctly applies the distributive property while carefully maintaining negative signs, and properly groups terms with identical variable combinations (a2a^2, abab, and b2b^2) to reach the final simplified form.

Step-by-Step Solution

1
Distribute −2a-2a into the first set of parentheses.
−6a2+8ab-6a^2 + 8ab
Applying the distributive property: (−2a×3a)+(−2a×−4b)(-2a \times 3a) + (-2a \times -4b). A negative term multiplied by a negative term results in a positive term.
2
Distribute −3b-3b into the second set of parentheses.
−3ab+6b2-3ab + 6b^2
Applying the distributive property: (−3b×a)+(−3b×−2b)(-3b \times a) + (-3b \times -2b). Again, the two negative factors produce a positive result.
3
Write out the full expanded expression by combining the distributed parts with the remaining term.
−6a2+8ab−3ab+6b2+5a2-6a^2 + 8ab - 3ab + 6b^2 + 5a^2
Setting up the expression to easily identify and group like terms together.
4
Group and combine the like terms.
−a2+5ab+6b2-a^2 + 5ab + 6b^2
Combining the a2a^2 terms (−6a2+5a2=−a2-6a^2 + 5a^2 = -a^2) and the abab terms (8ab−3ab=5ab8ab - 3ab = 5ab). The b2b^2 term remains unchanged.

Key Concept

Simplifying Expressions using the Distributive Property
Estimated Time:1m 0s
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Topics

Simplifying Algebraic Expressions

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