Question

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

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Answer 1

EXPLANATION.

α,β are the zeroes of the polynomial,

⇒ F(x) = x² - x - k = 0.

As we know that,

Sum of zeroes of the quadratic equation,

⇒ α + β = -b/a.

⇒ α + β = -(-1) ⇒ 1.

Products of zeroes of quadratic equation,

⇒ αβ = c/a.

⇒ αβ = -k.

⇒ α - β = 9 ⇒ (2).

From equation (1) and (2) we get,

⇒ α + β = 1.

⇒ α - β = 9.

We get,

⇒ 2α = 10.

⇒ α = 5.

Put the value of α = 5 in equation (1) we get,

⇒ 5 + β = 1.

⇒ α = -4.

Value of α = 5  and  β = -4.

⇒ αβ = -k.

⇒ (5)(-4) = -k.

⇒ -20 = -k.

⇒ k = 20

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.