# If α and β are the Zeros of the Polynomial f(x)=6x^2+x−2, Find the Value of (α/β+β/α)

By BYJU'S Exam Prep

Updated on: October 17th, 2023

If α and β are the zeros of the polynomial f(x)=6x2+x−2, find the value of (α/β+β/α)

In order to find the value of (α/β+β/α), we will require the values of α and β, which are the zeros of the polynomial. The zeros of f(x) are the values of x for which f(x) = 0. We can solve this by factoring, using the quadratic formula, or by completing the square.

After getting the values of α and β, we will use substitution method to put the values in (α/β+β/α) and get the desried result.

## If α and β are the Zeros of the Polynomial f(x)=6x2+x−2, Find the Value of (α/β+β/α)

Solution

To find the value of (α/β+β/α), we need to determine the values of α and β.

The zeros of the polynomial f(x) = 6x2 + x – 2 are the values of x for which f(x) = 0.

So, let’s solve the equation:

6x2 + x – 2 = 0

x = (-b ± √(b2 – 4ac)) / (2a)

For the given equation, a = 6, b = 1, and c = -2.

Using the quadratic formula, we have:

x = (-1 ± √(12 – 4(6)(-2))) / (2(6)) = (-1 ± √(1 + 48)) / 12 = (-1 ± √49) / 12 = (-1 ± 7) / 12

Therefore, we have two solutions:

x1 = (-1 + 7) / 12 = 6/12 = 1/2

x2 = (-1 – 7) / 12 = -8/12 = -2/3

Thus, α = 1/2 and β = -2/3.

Now, we can calculate (α/β+β/α):

(α/β + β/α) = (1/2) / (-2/3) + (-2/3) / (1/2) = (1/2) * (-3/2) + (-2/3) * (2/1) = -3/4 + (-4/3) = -9/12 + (-16/12) = (-9 – 16) / 12 = -25/12

Therefore, the value of (α/β+β/α) is -25/12.