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Answers
Question
Note :-
Here alpha = a , beta = b
If a,b are the zeroes of the polynomial f(x) = 2x²+5x+k satisfying the relations a²+b²+ab = 21/4,then find the value of k for this to be possible .
ANSWER
Given : -
If a,b are the zeroes of the polynomial f(x) = 2x²+5x+k satisfying the relations a²+b²+ab = 21/4.
Required to find : -
- Possible value of k ?
Conditions used : -
Here, conditions refer to the relations which are been used !
The relation between the sum of the zeroes & the coefficients is ;
✮ a + b = [ - (coefficient of x)]/[coefficient of x²]
Similarly,
The relation between the product of the zeroes & the coefficients is;
✮ ab = [constant term]/[coefficient of x²]
Solution : -
If a,b are the zeroes of the polynomial f(x) = 2x²+5x+k satisfying the relations a²+b²+ab = 21/4.
We need to find the value of k ?
Consider the given polynomial;
f(x) = 2x²+5x+k
The standard form of the polynomial is ;
- ax²+bx+c = 0
Compare the given polynomial with the standard form
2x²+5x+k = ax²+bx+c=0
Here,
- a = 2
- b = 5
- c = k
We know that;
The relation between the sum of the zeroes & the coefficients is;
a+b = [ - (coefficient of x)]/[coefficient of x²]
a+b = (-b)/(a)
a+b = (-5)/(2)
➝ a+b = -5/2 ............{1} equation-1
Similarly,
The relation between the product of the zeroes & the coefficients is;
ab = (constant term)/(coefficient of x²)
ab = (c)/(a)
ab = (k)/(2)
➝ ab = k/2 ................{2} equation-2
Consider Equation 1
a+b = -5/2
squaring on both sides
(a+b)² = (-5/2)²
a²+b²+2ab = 25/4
a²+b² +2(k/2) = 25/4 [from equation-2]
a²+b²+k = 25/4
a²+b² = 25/4 - k/1
➝ a²+b² = (25-4k)/(4) .............{3} equation-3
According to problem;
a²+b²+ab =21/4
➝ (25-4k)/(4)+(k)/(2) = 21/4 [from equation 2 & 3]
➝ (25-4k)/(4) = 21/4 - k/2
➝ (25-4k)/(4) = (21-2k)/(4)
4 get's cancelled in both LHS & RHS
➝ 25-4k = 21-2k
➝ 25-21 = -2k+4k
➝ 4 = 2k
➝ 2k = 4
➝ k = 4/2
➝ k = 2
Therefore,
value of k = 2
hope it will help you ❤❤
