If a² − b² = 9 and (a − b)² = 3 then ab is equal to

Explanation

We are given:

1. a² − b² = 9  

2. (a − b)² = 3

We can use the identity:

a² − b² = (a − b)(a + b)  

From (1):  

⇒ (a − b)(a + b) = 9 — (i)

From (2):  

(a − b)² = 3  

⇒ a − b = √3 or −√3 — (ii)

Substitute (ii) into (i):  

If a − b = √3  

Then (√3)(a + b) = 9  

⇒ a + b = 9 / √3 = 3√3

Now use the identity:  

(a + b)² = a² + 2ab + b²  

(a − b)² = a² − 2ab + b² = 3  

Add both:

(a + b)² + (a − b)² = 2a² + 2b² = 2(a² + b²)  

⇒ (a + b)² + 3 = 2(a² + b²)

We also know:  

a² − b² = 9, and  

(a + b)(a − b) = 9  

So, use formulas to isolate ab:

From identity:  

a² + b² = (a + b)² − 2ab  

a² − b² = 9 — already known

Now subtract equations:  

(a + b)² − (a − b)² = 4ab  

Use known values:  

(3√3)² − 3 = 4ab  

9×3 − 3 = 4ab  

27 − 3 = 4ab  

24 = 4ab  

⇒ ab = 6

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