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Difficulty: HardSimplifying Algebraic Expressions

Which of the following is the completely simplified form of the algebraic expression below?

4a(2a−3b)−[3b(a−4b)−2a(a+b)]4a(2a - 3b) - [3b(a - 4b) - 2a(a + b)]
  1. 10a2−13ab+12b210a^2 - 13ab + 12b^2Answer
  2. B
    6a2−17ab+12b26a^2 - 17ab + 12b^2
  3. C
    9ab9ab
  4. D
    6a2−13ab−12b26a^2 - 13ab - 12b^2

Answer

The simplified expression is 10a2−13ab+12b210a^2 - 13ab + 12b^2.
The correct answer is derived by first expanding the inner products (3ab−12b23ab - 12b^2 and −2a2−2ab-2a^2 - 2ab), combining them to get ab−12b2−2a2ab - 12b^2 - 2a^2, and then subtracting this entire grouped result from the expanded outer product (8a2−12ab8a^2 - 12ab). Properly distributing the negative sign to all terms from the bracket yields 10a2−13ab+12b210a^2 - 13ab + 12b^2.

Step-by-Step Solution

1
Distribute the terms inside the square brackets.
3b(a−4b)=3ab−12b23b(a - 4b) = 3ab - 12b^2 and −2a(a+b)=−2a2−2ab-2a(a + b) = -2a^2 - 2ab
According to the order of operations, simplify the innermost expressions first.
2
Combine like terms inside the brackets.
[3ab−12b2−2a2−2ab]=[ab−12b2−2a2][3ab - 12b^2 - 2a^2 - 2ab] = [ab - 12b^2 - 2a^2]
Simplifying the bracket's contents reduces complexity before distributing the negative sign.
3
Distribute the terms outside the brackets.
4a(2a−3b)=8a2−12ab4a(2a - 3b) = 8a^2 - 12ab
Expand the remaining multiplication.
4
Distribute the negative sign to the terms from the bracket.
−(ab−12b2−2a2)=−ab+12b2+2a2-(ab - 12b^2 - 2a^2) = -ab + 12b^2 + 2a^2
Subtracting a polynomial requires reversing the sign of every term inside.
5
Combine all remaining like terms.
8a2−12ab−ab+12b2+2a2=10a2−13ab+12b28a^2 - 12ab - ab + 12b^2 + 2a^2 = 10a^2 - 13ab + 12b^2
Group terms with identical variable parts to find the final simplified form.

Key Concept

Simplifying complex expressions by adhering strictly to the order of operations, accurately distributing terms and negative signs, and properly combining like terms.
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Topics

Simplifying Algebraic Expressions

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