Alpha , beta are zeroes of x square -3x -1 then find 1/ alpha + 1/ beta

Answer 1

To find 1/alpha + 1/beta, we need to first find the values of alpha and beta.

Given that alpha and beta are the zeroes of x^2 - 3x - 1 = 0, we can use the formula for finding the zeroes of a quadratic equation:

alpha, beta = (-b ± √(b^2 - 4ac))/(2a)

Here, the coefficients are:
a = 1
b = -3
c = -1

Substituting the values into the formula, we get:

alpha, beta = (-(-3) ± √((-3)^2 - 4(1)(-1)))/(2(1))
= (3 ± √(9 + 4))/2
= (3 ± √13)/2

So, alpha = (3 + √13)/2 and beta = (3 - √13)/2.

Now, let's find 1/alpha + 1/beta:

1/alpha + 1/beta = [(1*beta + 1*alpha)/(alpha*beta)]
= (alpha + beta)/(alpha*beta)

Using the sum and product of the roots formula for a quadratic equation:

alpha + beta = -(-3) = 3
alpha * beta = -1/1 = -1

Thus, 1/alpha + 1/beta = (alpha + beta)/(alpha * beta)
= 3/-1
= -3

Therefore, 1/ alpha + 1/ beta = -3.