Question

Find k if zeroes ∝ , β of the polynomial 5x² + (2k + 1)x + (k - 2) are such that 2∝ + 5β = 1.​

Answer 1

Given :

α and β are the zeroes of the given polynomial.

Now,

α + β = -b/a

α + β = - (2k + 1)/5

Multiplying both sides by 5,

5α + 5β = -(2k + 1).. (1)

Now,

αβ = c/a

αβ = (k - 2) / 5.. (2)

ATQ,

2α + 5β = 1

5β = 1 - 2α.. (3)

Putting this in eq. (1)

=> 5α + (1 - 2α) = -(2k + 1)

=> 3α = -2k - 2

=> α = (-2k - 2)/3.. (4)

Now,

Putting the value of α in eq. (3)

=> 5β = 1 - 2α

=> 5β = 1 - 2( -2k - 2 ) / 3

=> 5β = 1 + (4k + 4 / 3)

=> 5β = (4k + 4 + 3)/3

=> 5β = (4k + 7)/3

=> β = (4k + 7)/15.. (5)

Now,

From eq. (2), (4) and (5)

eq.(2) = eq. (4) × eq. (5)

(k - 2) / 5 = (-2k - 2)/3 × (4k + 7)/15

Simplifying this..

=> (k - 2) = -8k² - 14k - 8k - 14 / 3 × 3

=> 9k - 18 = -8k² - 22k - 14

=> -8k² - 22k - 14 + 18 - 9k = 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

Thus,

Value of k is 1/8 or -4.

Answer 2

Solution:-

Given:-

• Zeroes of Polynomial :- α and β.

To Find:-

• Value of k = ?

Find:-

• Sum of Zeroes = -b/a

=) α + β = - ( 2k +1)/5

=) 5 ( α + β ) = - ( 2k + 1)

=) 5β = - ( 2k + 1) - 5α _______(1)

• Product of Zeroes = c/a

=) αβ = ( k - 2)/5

=) 5 αβ = k - 2 ______________(2)

Now,

• 2α + 5β = 1 ( Given )

=) 5β = 1 - 2α_____________(3)

Now,

1 - 2α = - ( 2k + 1) - 5α [ ∵ L.H.S. are equal. ]

=) 5α - 2α = - 2k - 1 - 1

=) 3α = -2k - 2

=) α = ( -2k - 2)/3

Substituting [ α = ( -2k - 2)/3 ] in eq.3

=) 5β = 1 - 2α

=) 5β = 1 - 2[ ( -2k - 2)/3 ]

=) 5β = [ 3 - 2( -2k - 2)]/3

=) β = [ 3 + 4k + 4]/15

Substituting the value of α and β in eq. 2

=) 5 αβ = k - 2

=) 5 [ ( -2k - 2)/3 × (3 + 4k + 4)/15 ] = k - 2

=) ( -2k - 2)/3 × (3 + 4k + 4)/15 = ( k -2)/5

=) ( -2k - 2)/3 × (3 + 4k + 4)/3 = ( k -2)

=) [ (-2k -2 )(4k + 7) ]/9 = ( k-2)

=) [ -8k² - 14k - 8k - 14 ) = 9k - 18

=) -8k² - 14k - 8k - 14 - 9k + 18 = 0

=) -8k² - 31k + 4 = 0

=) 8k² + 31k - 4 = 0

=) 8k² + ( 32k - k ) - 4 = 0

=) 8k² + 32k - k - 4 = 0

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

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

=) [ k = -4 ] and [ k = 1/8 ]

Hence,

Value of k is -4 and 1/8.