10. In polynomial X^2 - kx + 6 The zeroes of are in the ratio 3 : 2 , find k.
Answers
Step-by-step explanation:
Given :-
In polynomial X^2 - kx + 6 The zeroes of are in the ratio 3 : 2.
To find :-
Find the value of k ?
Solution :-
Given quadratic polynomial X²-kX+6
On comparing this with the standard quadratic polynomial ax²+bx+c
We have ,
a = 1
b = -k
c = 6
Ratio of the zeroes of the given Polynomial = 3:2
Let they be 3A and 2A
We know that
The sum of the zeores = -b/a
=> 3A + 2A = -(-k)/1
=> 5A = k/1
=> 5A = k
=> k = 5A -------(1)
The Product of the zeroes = c/a
=> (3A)(2A) = 6/1
=> 6A² = 6
=> A² = 6/6
=> A² = 1
=> A = ±√1
=> A =± 1
=> A = 1 or -1 ----------(2)
On Substituting the value of A in (1) then
If A = 1 then
=> k = 5(1)
=> k = 5
If A = -1 then
=> k = 5(-1)
=> k = -5
Therefore,k = 5 or -5
Answer:-
The value of k for the given problem is 5 or -5
Check:-
If k = 5 then the Polynomial will be x²-5x+6
=> x²-5x+6 = 0
=> x²-2x-3x+6 = 0
=> x(x-2)-3(x-2) = 0
=> (x-2)(x-3) = 0
=> x-2 = 0 or x-3 = 0
=> x = 2 or 3
Zeroes are 2 and 3
Their ratio = 3:2
If k = -5 then the Polynomial will be x²+5x+6
=> x²+5x+6 = 0
=> x²+2x+3x+6 = 0
=> x(x+2)+3(x+2) = 0
=> (x+2)(x+3) = 0
=> x+2 = 0 or x+3 = 0
=> x = -2 or -3
Zeroes are -2 and -3
Their ratio = -3:-2 = -3/-2 = 3/2 = 3:2
Verified the given relations in the given problem
Used formulae:-
- The standard quadratic polynomial is ax²+bx+c
- Sum of the zeroes = -b/a
- Product of the zeroes = c/a
- To get the zeores of a polynomial we equate it to zero.