Question

If alpha and beta are zeroes of polynomial f(x)=x2-x-k such that alpha -beta =9 , find k

Answer 1

f(x)=x²-x-k=0, α ,β are the zeroes of the polynomial f(x)=x²-x-k=0
then
a=1,b=1 and c=k
α+β=-(b/a)=-1. -1. (sum of roots)
α-β=9. -2
adding 1 & 2
2α=8
α=4
putting α=4 in 1
4+β=-1
β=-5


therefore α=4,β=-1
now αβ=c/a. (product of roots)
4*(-1)=k/1
therefore k=-4

Answer 2

Gɪᴠᴇɴ :

α & β are zeroes of polynomial f(x).

Where,

f(x) = x² - x - k

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

➣ The general form of an quadratic polynomial is,

$\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{x^2\:-\:(\alpha\:+\:\beta)\:x\:+\:\alpha\:\beta}}}}}}$

➣ Compare given quadratic polynomial to the general form of quadratic equation, we get

➾ α + β = 1 ----(a)

Aɴᴅ,

➾ αβ = -k ----(i)

Aᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ᴛʜᴇ ǫᴜᴇsᴛɪᴏɴ,

➾ α - β = 9 ----(b)

➣ Now adding equation (a) & (b), we get

⇒ α + β + α - β = 1 + 9

⇒ 2α = 10

⇒ α = $\rm{\dfrac{10}{2}}$

⇒ $\bf\orange{\alpha\:=\:5}$

➣ Putting the value of α in equation (a), we get

➾ α + β = 1

➾ 5 + β = 1

➾ β = 1 - 5

➾ $\bf\blue{\beta\:=\:-4}$

➣ Let us putting the value of α & β in equation (i), we get

$:\implies$ 5 × (-4) = -k

$:\implies$ -20 = -k

$:\implies$ ${\underline{\red{\boxed{\bf{\pink{k\:=\:20}}}}}}\:\green\bigstar$

$\Large\bf\purple{Therefore,}$

The value of 'k' is 20.