Question

if one zero of the quadratic polynomial 4x^2 - 10x -(k2 + 4k) is reciprocal of the other zero, then find the value of k

Answer 1

Given :-

One zero of the quadratic polynomial 4x² - 10x - ( k² + 4k ) is reciprocal of the other zero .

To Find :-

The Value of "k"

Used Concepts :-

For quadratic polynomial " ax² + bx + c " :-

Sum of zeroes = $\dfrac{-b}{a}$

Product of zeroes = $\dfrac{c}{a}$

Solution :-

Let , p ( x ) = 4x² - 10x - ( k² + 4k )

1st zero of p ( x ) = λ

Then , According to Question ;

Second zero of p ( x ) = $\dfrac{1}{λ}$

On comparing p ( x ) with general form of a quadratic polynomial " ax² + bx + c " we get ;

a = 4

b = - 10

c = - ( k² + 4k )

As we knows that , Product of roots = c/a

=> $λ × \dfrac{1}{λ} = \dfrac{c}{a}$

=> $λ × \dfrac{1}{λ} = \dfrac{-(k² + 4k)}{4}$

=> $1 = \dfrac{-k²-4k}{4}$

=> $-k² - 4k = 4$

=> $-k² - 4k - 4 = 0$

=> Applying Splitting the middle term we get ;

=> $-k² - 2k - 2k - 4 = 0$

=> $-k ( k + 2 ) - 2 ( k + 2 ) = 0$

=> $( k + 2 ) ( - k - 2 ) = 0$

=> Either , k + 2 = 0 or - k - 2 = 0

=> k = - 2 or - k = 2

=> k = - 2 or k = - 2

Here , both values of k are same .

Henceforth , Value of k is " - 2 " .