Question

If one zero of the polynomial 2x

2 – 5 x – (a – 4) is the reciprocal of the other, find the

value of ‘a’

Answer 1

The given polynomial is

f (x) = 2x² - 5x - (a - 4)

We can take α, 1/α to be the zeroes of f (x).

By the relation between zeroes and coefficients, we can write

α + 1/α = - (- 5)/2 .....(1)

α * 1/α = - (a - 4)/2 .....(2)

We take the (2) no. relation:

α * 1/α = - (a - 4)/2

or, 1 = - (a - 4)/2

or, - (a - 4)/2 = 1

or, a - 4 = - 2

or, a = 4 - 2

or, a = 2

Therefore the value of a is 2.

Answer 2

The value of 'a' is 2

Given:

2x² - 5x - (a - 4)

Step-by-step explanation:

The zeroes of the given polynomial is:

α and 1/α

Now, the product of the roots is given as:

α × 1/α = c/a = - (a - 4)/2 = (4 - a)/2 → (equation 1)

The sum of the roots is given as:

α + 1/α = - b/a = - (- 5)/2 = 5/2 → (equation 2)

Now, on taking the equation (1), we get,

α × 1/α = (4 - a)/2

1 = (4 - a)/2

2 = 4 - a

a = 4 - 2

∴ a = 2