if alpha and beta are the zeroes of 2x^2+5x+k and alpha ^2+beta^2+alphabeta=21/4 fond the values of k.anu says that the zeroes of the given polynomial are reciprocal to each other.is she right?justify
Answers
Answer:
♦ k = 2
♦ Yes , it is right to say that the zeros of the given quadratic polynomial are reciprocal of each other .
Note:
★ The possible values of the variable for which the polynomial becomes zero are called its zeros .
★ A quadratic polynomial can have atmost two zeros .
★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;
• Sum of zeros , (α + ß) = -b/a
• Product of zeros , (αß) = c/a
Solution:
Here,
The given quadratic polynomial is ;
2x² + 5x + k .
On comparing with the general form of a quadratic polynomial ax² + bx + c ,
We have ;
a = 2
b = 5
c = k
Also,
It is given that , α and ß are the zeros of the given quadratic polynomial .
Thus,
=> Sum of zeros = -b/a
=> α + ß = -5/2
Also,
=> Product of zeros = c/a
=> αß = k/2
Now,
According to the question , we have ;
=> α² + ß² + αß = 21/4
=> α² + 2αß + ß² - αß = 21/4
=> (α + ß)² - αß = 21/4
=> (-5/2)² - k/2 = 21/4
=> 25/4 - k/2 = 21/4
=> k/2 = 25/4 - 21/4
=> k/2 = (25 - 21)/4
=> k/2 = 4/4
=> k/2 = 1
=> k = 1×2
=> k = 2
Hence , k = 2
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Again,
=> Product of zeros = c/a
=> αß = k/2
=> αß = 2/2
=> αß = 1
=> α = 1/ß or ß = 1/α