If 𝑎2+2𝑎𝑏−3𝑏2−41=0a 2 +2ab−3b 2 −41=0, then 𝑎2−𝑏2=?a 2 −b 2 =?

The given equation is a quadratic equation in terms of 'a'. We can rewrite it as:

a² + 2ab - 3b² = 41

We are asked to find the value of a² - b². We know that a² - b² can be written as (a+b)(a-b).

We already have a term in the given equation that looks like (a+b), which is a + √3b.

So, we can rewrite the given equation in terms of (a+b) as follows:

(a + √3b)² = 41

Taking square root on both sides, we get:

a + √3b = √41

Now, we can find the value of (a-b) from the above equation.

a - √3b = -√41

Now, we can find the value of (a² - b²) by multiplying (a+b) and (a-b):

(a² - b²) = (a + √3b) * (a - √3b) = √41 * -√41 = -41

So, a² - b² = -41.