If (𝑎 + 𝑏)2 = 10 , (𝑎 − 𝑏)2 = 6 , find(1) 𝑎2 + 𝑏2(2) 𝑎𝑏(3) 2𝑎2 + 7𝑎𝑏 + 2𝑏2

Sure, let's solve this step by step:

(1) To find 𝑎² + 𝑏², we can use the formula (𝑎 + 𝑏)² = 𝑎² + 2𝑎𝑏 + 𝑏² and (𝑎 - 𝑏)² = 𝑎² - 2𝑎𝑏 + 𝑏².

Adding these two equations, we get:

(𝑎 + 𝑏)² + (𝑎 - 𝑏)² = 2𝑎² + 2𝑏²

Substituting the given values, we get:

10 + 6 = 2𝑎² + 2𝑏²

16 = 2𝑎² + 2𝑏²

So, 𝑎² + 𝑏² = 16/2 = 8

(2) To find 𝑎𝑏, we can subtract the second equation from the first:

(𝑎 + 𝑏)² - (𝑎 - 𝑏)² = 4𝑎𝑏

Substituting the given values, we get:

10 - 6 = 4𝑎𝑏

4 = 4𝑎𝑏

So, 𝑎𝑏 = 4/4 = 1

(3) To find 2𝑎² + 7𝑎𝑏 + 2𝑏², we can substitute the values of 𝑎² + 𝑏² and 𝑎𝑏 that we found:

2(8) + 7(1) + 2(8) = 16 + 7 + 16 = 39