What can you say about a and b if (a + b)^2 < a^2 + b^2? — Mathematics Part 1

Question 2 of 43intermediate🔍 AnalyzeShort Answer2 marks

What can you say about a and b if (a + b)^2 < $a^{2}$ + $b^{2}$ ?

Correct Answer

a and b must have opposite signs (i.e., their product ab is negative, meaning one is positive and the other is negative).

Exercise: Think and Reflect | Q: 1 | (Page 4)

For More Understanding

Explanation

The textbook context provides the identity (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$ and prompts the student to use the expansion to compare (a + b)^2 and $a^{2}$ + $b^{2}$ . By substituting the expansion into the inequality (a + b)^2 < $a^{2}$ + $b^{2}$ , we get $a^{2}$ + 2ab + $b^{2}$ < $a^{2}$ + $b^{2}$ , which simplifies to 2ab < 0. Thus, ab < 0, meaning a and b have opposite signs.

Solution Steps

Step 1: Write the identity (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$ .
Step 2: Substitute the expansion into the inequality: $a^{2}$ + 2ab + $b^{2}$ < $a^{2}$ + $b^{2}$ .
Step 3: Simplify the inequality by subtracting $a^{2}$ + $b^{2}$ from both sides to get 2ab < 0, or ab < 0.
Step 4: Conclude that since ab < 0, a and b must have opposite signs (one positive, one negative).

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