Question

If Alrha And beta are the zeroes of the polynomial f(x) = x2 - 5x-k such that alpha-beta= 1 find value of k

Answer 1

f(x)= x2-5x+k

Here, a=1, b=-5 and c=k

Now, alpha+beta= -b/a= -(-5)/1= 5

alpha*beta= c/a= k/1= k

Now, apha-beta=1

Squaring both sides, we get,

(alpha-beta)2=12

=> (alpha+beta)2-4*aplha*beta=1

=> (5)2-4k=1

=> -4k= 1-25

=> -4k= -24

=> k=6

Answer 2

Answer:

k = 6

Step-by-step explanation:

Given :-

Alpha and Beta are the zeroes of the polynomial x² - 5x - k.

Alpha - Beta = 1

To find :-

We need to find the value of k.

Solution :-

Compare the given quadratic polynomial with the general form.

General form :-

• ax² + bx + c = 0

On comparing ,

• a = 1

• b = - 5

• c = k

Sum of roots, α + β = $\bf\frac{-b}{a}$

α + β = $\bf\frac{-(-5)}{1}$

α + β = $\bf\frac{5}{1}$

α + β = 5 ----> (1)

Product of roots, αβ = $\bf\frac{c}{a}$

αβ = $\bf\frac{k}{1}$

αβ = k ----> (2)

Alpha - Beta = 1

α - β = 1 -----> (3)

Adding equation 1 and equation 3,

α + β = 5 ----> (1)

α - β = 1 ----> (2)

----------------

2α = 6

=> α = $\bf\frac{6}{2}$

=> α = 3

Substitute α = 3 in equation 3,

=> α - β = 1

=> 3 - β = 1

=> 3 - 1 = β

=> 2 = β

Substitute α = 3 and β = 2 in equation 2,

=> αβ = k

=> 3 × 2 = k

=> 6 = k

•°• Value of k = 6