Question 6 of 43beginner💡 UnderstandShort Answer2 marks

What if we replace b by -b in (a + b)^2 = a2a^{2} + 2ab + b2b^{2}?

Correct Answer

If we replace b by -b in the identity (a + b)^2 = a2a^{2} + 2ab + b2b^{2}, we obtain the expansion for (a - b)^2.

By substituting -b in place of b, the left side becomes [a + (-b)]^2, which is (a - b)^2. On the right side, the terms become a2a^{2} + 2a(-b) + (-b)^2.

Simplifying this, we get a2a^{2} - 2ab + b2b^{2}. Therefore, the new identity formed is (a - b)^2 = a2a^{2} - 2ab + b2b^{2}.

Exercise: Think and Reflect | Q: 1 | (Page 6)
For More Understanding

Explanation

The textbook demonstrates how replacing b with -b in an identity yields a new identity, such as deriving (a-b)^3 from (a+b)^3. Applying this same substitution to (a+b)^2 = a2a^{2} + 2ab + b2b^{2} results in (a-b)^2 = a2a^{2} - 2ab + b2b^{2}, which is also explicitly derived from areas in the context.

Solution Steps

  1. Step 1: Replace b with -b in the identity: [a + (-b)]^2 = a2a^{2} + 2a(-b) + (-b)^2

  2. Step 2: Simplify the left side: (a - b)^2

  3. Step 3: Simplify the right side: a2a^{2} - 2ab + b2b^{2}

  4. Step 4: Conclude the new identity: (a - b)^2 = a2a^{2} - 2ab + b2b^{2}

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