Question

Find k if zeroes alpha and beta of the polynomial 5x^2+(2k+1)x+(k-2) are such that 2alpha +5beta=1

Answer 1

heya...!!

given that;-

alpha and beta are the zeroes of the polynomial

now,

alpha + beta = -(2k +1)/5

=> 5alpha + 5beta = -(2k +1)--------(2)

alpha × beta = (k -2)/5------(2)

given that:-

2alpha + 5beta = 1

5beta = (1 - 2alpha)----(3) put in--(1)

we get,

5alpha + 1 - 2alpha = -(2k +1)

=> 3alpha = -2k - 2

=> alpha = (-2k - 2)/3-----(4) put in --(3)

we get,

5beta = 1 - 2(-2k - 2)/3

=> 5beta = (3 + 4k + 4)/3

=> beta = (4k + 7)/15---(5)


from---(2) ,(4) and (5)

we get,


(-2k - 2)/3 × (4k + 7)/15 = (k - 2)/5

=> (-8k² -14k - 8k - 14)/3×3 = (k -2)

=> -8k² - 22k - 14 = 9k - 18

=> -8k² - 22k - 9k - 14 + 18 = 0

=> -8k² - 31k + 4 = 0

=> 8k² + 31k - 4 = 0

=> 8k² + 32k - k - 4 = 0

=> 8k(k + 4) - 1(k + 4) = 0

=> (8k -1)(k + 4) = 0

=> k = 1/8 or k = -4

hepe it's help you

Answer 2

I hope this helps you.

: )