Question

if alpha and beta are the zeros of quadratic polynomial x square - 6 X + a find the value of 'a' 3 alpha + 2 Beta equals to 20

Answer 1

Then α+β=6α+β=6 and αβ=kαβ=k

As per given condition, 3α+2β=203α+2β=20

Solving these simultaneous equations, we get α=8α=8 and β=−2β=−2

Hence, k=αβ=−16

The given quadratic equation is

x2−6x+k=0x2−6x+k=0 … (1)

A quadratic equation is always of the form

x2−(Sumofroots)x+(Productofroots)=0x2−(Sumofroots)x+(Productofroots)=0 … (2)

Here, the roots are given as ααand ββ.

From (1) & (2), we get,

Sum of roots = α+β=6α+β=6 … (3)

Product of roots = αβ=kαβ=k

From given data,

3α−2β=203α−2β=20 … (4)

Solving equations (3) & (4), we get,

(3) × 2 => 22α+2β=12α+2β=12

(4) => 3α−2β=203α−2β=20

=>5α=32=>5α=32 [By cancelling 2β2β in both equations.]

=>α=325=>α=325

Substitute ααvalue in equation (3).

325+β=6325+β=6

β=6−325β=6−325

β=(30−32)5β=(30−32)5

β=−25β=−25

Thus, α=325,β=−25α=325,β=−25

Now, k=αβk=αβ

Therefore, k=325×−25k=325×−25

k=−64/25