Math Problem Statement

(a - 3b)^2 - (a + b)^2

Solution

Let's solve the given expression:

(a3b)2(a+b)2(a - 3b)^2 - (a + b)^2

Step 1: Expand both squares

For (a3b)2(a - 3b)^2, use the expansion formula (xy)2=x22xy+y2(x - y)^2 = x^2 - 2xy + y^2:

(a3b)2=a22a(3b)+(3b)2=a26ab+9b2(a - 3b)^2 = a^2 - 2a(3b) + (3b)^2 = a^2 - 6ab + 9b^2

For (a+b)2(a + b)^2, use the expansion formula (x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2:

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Step 2: Subtract the two expanded forms

Now, subtract (a+b)2(a + b)^2 from (a3b)2(a - 3b)^2:

(a3b)2(a+b)2=(a26ab+9b2)(a2+2ab+b2)(a - 3b)^2 - (a + b)^2 = \left( a^2 - 6ab + 9b^2 \right) - \left( a^2 + 2ab + b^2 \right)

Distribute the negative sign:

=a26ab+9b2a22abb2= a^2 - 6ab + 9b^2 - a^2 - 2ab - b^2

Step 3: Combine like terms

Cancel out a2a^2 and combine the remaining terms:

=(6ab2ab)+(9b2b2)= (-6ab - 2ab) + (9b^2 - b^2)

=8ab+8b2= -8ab + 8b^2

Step 4: Final Answer

Thus, the simplified expression is:

8b28ab8b^2 - 8ab

This matches option A in the provided choices.

Would you like further details on any step?

Here are some related questions to practice:

  1. How do you expand binomials using the square of a sum or difference formula?
  2. What is the general formula for (xy)2(x - y)^2 and (x+y)2(x + y)^2?
  3. How do you simplify expressions with like terms?
  4. Can you expand and simplify (2x5y)2(x+4y)2(2x - 5y)^2 - (x + 4y)^2?
  5. How does changing the signs of terms in a binomial affect the result?

Tip: When simplifying expressions involving squares of binomials, always expand each term separately before combining like terms.

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Math Problem Analysis

Mathematical Concepts

Algebra
Binomial Expansion
Difference of Squares

Formulas

(x - y)^2 = x^2 - 2xy + y^2
(x + y)^2 = x^2 + 2xy + y^2

Theorems

Binomial Expansion Theorem

Suitable Grade Level

Grades 8-10