Question

please can anyone answer the question in the attachment

Answer 1

Answer:

( - ∞ , 6/5 ] ∪ [ - 1 , ∞ )

Step-by-step explanation:

Equation is : 5x^2 - 4x + 2 + k( 4x^2 - 2x - 1 ) = 0

  ⇒ 5x^2 - 4x + 2 + 4kx^2 - 2kx - k = 0

  ⇒ 5x^2 + 4kx^2 - 4x - 2kx + 2 - k = 0

  ⇒ x^2( 5 + 4k ) - x( 4 + 2k ) + ( 2 - k ) = 0

 Here,

Roots of the given equation are real. It means its discriminant is greater than or equal to 0.

discriminant ≥ 0 { in equation ax^2 + bx + c = 0, discriminant is given by b^2 - 4ac }

  In question, discriminant is ( 4 + 2k )^2 - 4( 5 + 4k )( 2 - k ) ≥ 0

⇒ [ 16 + 4k^2 + 8k ] - 4[ 10 - 5k + 8k - 4k^2 ] ≥ 0

⇒ 16 + 4k^2 + 8k - 4[ 10 + 3k - 4k^2 ] ≥ 0

⇒ 16 + 4k^2 + 8k - 40 - 12k + 16k^2 ≥ 0

⇒ 4k^2 + 16k^2 + 8k - 12k + 16 - 40 ≥ 0

⇒ 20k^2 - 4k - 24 ≥ 0

⇒ 5k^2 - k - 6 ≥ 0

⇒ 5k^2 - ( 6 - 5 )k - 6 ≥ 0

⇒ 5k^2 + 5k - 6k - 6 ≥ 0

⇒ 5k( k + 1 ) - 6( k + 1 ) ≥ 0

⇒ ( 5k - 6 )( k + 1 ) ≥ 0

 solution set is ( - ∞ , 6/5 ] ∪ [ - 1 , ∞ )