Question

If 5a – b = 9 and ab = 2, find the value of 25a² + b².

Answer 1

Given: 5a – b = 9 and ab = 2

We need to find the value of 25a² + b²

Step 1: Express 25a² + b² in a factorable form

Note that 25a² + b² = (5a)² + b²

If we can find (5a – b)², this might help:

(5a – b)² = 25a² – 2 × 5a × b + b²

(5a – b)² = 25a² – 10ab + b²

Step 2: Use the given values

From the given,

5a – b = 9 

⇒ (5a – b)² = 9²

⇒ (5a – b)² = 81

And ab = 2

Substitute ab = 2 into the expanded expression:

81 = 25a² – 10 × 2 + b² 

81 = 25a² – 20 + b²

Step 3: Solve for 25a² + b²

25a² + b² = 81 + 20

25a² + b² = 101