Question

if alpha and beta are the zeros of quadratic polynomial f x is equal to x square - 5 x + K such that Alpha minus beta is equals to 1 find the value of k​

Answer 1

K = -5

Please note, Instead of alpha and beta, I've used a & b, respectively.

f(x) = x² - 5x + k

Let a, b be the zeroes of f(x)..

Given: a - b = 1

=> a = b + 1

Therefore, the zeroes are b, b + 1

Substituting these values in f(x), we get 2 equations:

i) (b)² - 5b + k = 0

ii) (b+1)² - 5(b+1) + k = 0

Subtract equation (i) from (ii),

[(b+1)² - b²] + [-5(b+1) -(-5b)] + [k - k] = 0

b² + 2b + 1 - b² - 5b + 5 + 5b = 0

2b + 1 + 5 = 0

2b = -6

b = -6/2 = -3

=> a = -3 + 1 = -2

We, know that product of the zeroes =

constant/(coefficient of x²)

=> a + b = k/1

=> -2 -3 = k

=> k = -5