Question

For what value of k, x² - 5x + 3(k -1) = 0 has difference of roots equal to 11

Answer 1

Let α  and  β are the roots of the quadratic equation.

Given quadratic equation is  x² - 5x + 3(k -1) = 0.
α -  β= 11 …………(1)

On comparing with ax² + bx + c
a= 1 , b= -5, c= 3(k-1)

Sum of zeroes (α+β) = -b/a
(α+β) = -b/a
(α+β) = -(-5)/1= 5
(α+β) = 5……………..(2)

On Adding Equations 1 and 2,
α -  β= 11
α + β = 5
---------------
2α = 16
α = 16/2
α = 8

On putting α = 8 in eq 1,
α -  β= 11
8 - β = 11
8-11 = β
β = -3

Product of zeroes(α.β)= c/a
8 × -3 =  3(k-1)/1           [α = 8 , β = -3]
-24 =  3(k-1)
-24 = 3k -3
-24 +3 = 3k
-21 = 3k
k = -21/3
k = -7

Hence, the value of k = -7.

HOPE THIS WILL HELP YOU…

Answer 2

Hey


The given equation is :-


x² - 5x +3(k - 1 ) = 0



Let the root be a and b respectively .


Now ,

given that difference of root = 11


Let a > b

So ,


a - b = 11 –––( i )


And ,

We know that ,


sum of roots = -( b ) / a


so , a + b = - ( -5 ) / 1


=> a + b = 5 ––– ( ii )


Now , adding eq ( i )and ( ii ) , we get


a - b + a + b = 11 + 5


=> 2a = 16


=> a = 8



So ,

8 - b = 11

=> - b = 3


=> b = -3 .



Now ,


We know that ,

product of zeros = c / a


So ,

a * b = c / a


=> 8 * ( - 3 ) = 3 ( k - 1 ) / 1



=> - 24 = 3 ( k - 1 )


=> -8 = k - 1


=> k - 1 = -8


=> k = -8 + 1


=> k = -7


So ,

required value of k = (-7)

You can check it by putting the value of k in the equation .



thanks :)