Question

find the value of k for which one root of the quadratic equation X square - 14 x + 8 = 0 is 6 times the other

Answer 1

Let roots be α and β
A/q
α = 6β 
now, if α and β are roots then equation will be (x -α)(x -β) =0
 (x -α)(x -β) =0
⇒ x² - (α+β)x + αβ =0 
now putting α = 6β ,
⇒x² - (6β +β)x + 6β×β =0
⇒x² - 7βx +6β² =0 
now comparing with  kx² -14x +8 =0
7β =14/k
⇒β =2/k
⇒β² = 4/k²_______(1)
and 6β² =8/k
⇒β² =4/3k_______(2)
equating (1) and (2), we get,
4/k² = 4/3k
⇒k =3

Answer 2

Let roots be α and β
Acc. to question
α = 6β 
now, if α and β are roots then equation will be (x -α)(x -β) =0
 (x -α)(x -β) =0
⇒ x² - (α+β)x + αβ =0 
now putting α = 6β ,
⇒x² - (6β +β)x + 6β×β =0
⇒x² - 7βx +6β² =0 
now comparing with  kx² -14x +8 =0
Using relationship
α+β= -(b)/a
7β =14/k
⇒β =2/k
⇒β² = 4/k²_______(1)
and 6β² =8/k
⇒β² =4/3k_______(2)
equating (1) and (2), we get,
4/k² = 4/3k
⇒k =3

Hope my answer will help you

MARK IT AS BRAINLIEST ✌️