Question 3 of 43intermediate🔍 AnalyzeShort Answer2 marks

What can you say about a and b if (a + b)^2 > a2a^{2} + b2b^{2}?

Correct Answer

Using the algebraic identity, we know that (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2.

Given the condition (a+b)2>a2+b2(a + b)^2 > a^2 + b^2, we can substitute the expansion into the inequality. This gives us a2+2ab+b2>a2+b2a^2 + 2ab + b^2 > a^2 + b^2.

Subtracting a2+b2a^2 + b^2 from both sides, we get 2ab>02ab > 0, which simplifies to ab>0ab > 0. Therefore, the product of aa and bb must be positive. This implies that aa and bb must have the same sign (both positive or both negative).

Exercise: Think and Reflect | Q: 2 | (Page 4)
For More Understanding

Explanation

The textbook prompts the student to use the expansion of (a+b)2(a + b)^2 to decide which term makes it larger. By substituting the identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 into the inequality (a+b)2>a2+b2(a + b)^2 > a^2 + b^2, the a2a^2 and b2b^2 terms cancel out, leaving 2ab>02ab > 0. Thus, the product abab must be positive, meaning both numbers share the same sign.

Solution Steps

  1. Step 1: Write the algebraic identity: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

  2. Step 2: Substitute the identity into the given inequality: a2+2ab+b2>a2+b2a^2 + 2ab + b^2 > a^2 + b^2

  3. Step 3: Subtract a2+b2a^2 + b^2 from both sides: 2ab>02ab > 0

  4. Step 4: Conclude that ab>0ab > 0, meaning aa and bb have the same sign.

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