What can you say about a and b if (a + b)² > a² + b²? Class 9

What can you say about a and b if (a + b)² > a² + b²? Class 9

Question 1.

What can you say about a and b if (a + b)² > a² + b²? Class 9

Answer:

If (a + b)² > a² + b², then

a² + 2ab + b² > a² + b² ⇒ 2ab > 0

So, ab is positive, meaning a and b have the same sign (both positive or both negative).

Question 2.

When will (a + b)² be equal to a² + b²?

Did you observe that (a + b)² and a²+ b² are both positive? What term will decide which is larger? Use the expansion of (a + b)² to decide. Class 9

Answer:

When (a + b)² = a² + b²,

a² + 2ab + b² = a² + b² ⇒ 2ab = 0 ⇒ ab = 0 So, at least one of a or b is zero.

Observation:

Since squares of real numbers are never negative, both (a + b)² and a² + b² are always non-negative.

The difference between the two expressions is (a + b)² - (a² + b²) = 2ab

So, the term 2ab decides which expression is larger:

i. positive → (a + b)² is larger

ii. negative → a² + b² is larger