a^2-2ab-15b^2
a^2-2ab-15b^2 =
(a - b)^2 - 16 b^2 =
[a - b + 4 b]* [a - b - 4b] =
(a + 3b)*(a - 5b)
To find the factorization of a^2-2ab-15b^2, we can use the quadratic formula. The quadratic formula states that for any quadratic equation in the form of ax^2 + bx + c = 0, the solutions for x can be found using the equation:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, we have a = 1, b = -2ab, and c = -15b^2. Plugging these values into the quadratic formula, we get:
x = (-(-2ab) ± √((-2ab)^2 - 4ac)) / 2(1)
x = (2ab ± √(4a^2b^2 - 4(-15b^2))) / 2
x = (2ab ± √(4a^2b^2 + 60b^2)) / 2
x = (2ab ± √(4b^2(a^2 + 15))) / 2
x = (2ab ± √(4b^2) √(a^2 + 15)) / 2
x = (2ab ± 2b √(a^2 + 15)) / 2
x = ab ± b √(a^2 + 15)
Now, since a^2-2ab-15b^2 = (a + 3b)(a - 5b), we can see that the factorization is (a + 3b)(a - 5b).