Question

If α and β are the zeroes of the polynomial x² - 5x + k and α - β = -1 , find the value of 'k'.

Answer with complete steps.

Points : 20☺

Answer 1

step1 :-
alpha + beta = -b/a = -(-5)/1 = 5
alpha.beta = K/1 = K


step 2 :- alpha - beta = -1
we know,
(a -b)² = ( a + b)² -4ab
use this

(alpha + beta )² -4alpha.beta = ( alpha -beta)²

now put step 1 results

( 5)² - 4( K) = (-1)²

25 - 4K = 1

24 -4K =0

K = 6

Answer 2

Let p(x)=x^2-5x+k
Given
Zeroes of p(x) are alpha and beta
And
Alpha - beta = -1-----(1)
Compare p(x) with ax^2+bx+c=0
a=1, b=-5, c=k
i) alpha +beta = -b/a=-(-5)/1=5---(2)
ii)alpha*beta=c/a= k/1---(3)
(alpha+beta)^2-(alpha-beta)^2=4*alpha*beta
5^2-(-1)^2=4*k
25-1=4k
24=4k
24/4=k
K=6