Question

if alpha and beta are the zeros of a polynomial x square - 5 x + k and alpha minus beta is equal to minus 1 find the value of k​

Answer 1

Given equation :- x² - 5x + k = 0

Where , a = 1 , b = - 5 , c = k

α and β are the zeros of the polynomial.

Then , Sum of the zeros ( α + β ) = c / a = k / 1 = k

And , Product of the zeros ( αβ ) = -b / a = - ( -5 ) / 1 = 5

Now given ,

α - β = - 1

( α - β )² = ( - 1 )² [ • Both sides square ]

α² - 2αβ + β² = 1 [ • Using identity , ( a - b )² = a² - 2ab + b² ]

( α² + β² ) - 2αβ = 1

( α + β )² - 2αβ - 2αβ = 1 [ • Using identity , a² + b² = ( a + b )² - 2ab ]

( k )² - 4 × 5 = 1 [ • Putting the values ]

k² - 20 = 1

k² = 1 + 20

k² = 21

k = √21 [ • Required Answer ]