If If α and β are the zeros of the quadratic polynomial p(x)= x2-5x+k such that α-β=1, find . . .

Question : If If α and β are the zeros of the quadratic polynomial p(x)= x²-5x+k such that α-β=1, find the value of k.

Doubt by Yathartha

Solution :  

p(x)= x²-5x+k

a = 1
b = -5
c = k

α+β = -b/a
α+β = -(-5)/1
α+β = 5 — (1)

αβ = c/a
αβ = k/1
αβ = k — (2)

Now 

α-β=1 (Given) — (3)

You can solve equation (1) and (3) by elimination method and you will get α=3 & β=2. 

Putting these values in equation (2)
you will get
αβ = k
(3)(2) = k
6=k
k=6

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Alternate Method

p(x)= x2-5x+k

a = 1
b = -5
c = k

α+β = -b/a
α+β = -(-5)/1
α+β = 5 

αβ = c/a
αβ = k/1
αβ = k 

(α−β)² = (α+β)² - 4αβ
(1)² = (5)² - 4(k)
4k = 25-1
4k = 24
k = 24/4
k = 6