Question

If α and β are the zeros of quadratic polynomial x²-5x²+6. Then find a quadratic polynomial whose roots are 2α and 2β.

Answer 1

Using Quadratic Formula- x²-(sum of zeroes)x + (products of zeroes)

two zeroes are given 2alpha & 2 beta

therefore,

alpha=1/2

beta=1/2

putting alpha and beta in the formula

x²-(sum of zeroes)x + (products of zeroes)

x²-(1/2+1/2)x + (1/2*1/2)

solves this u will get the equation

4x²-4x+1

Answer 2

Solution :-

It is given that α and β are the zeros of polynomial f(x) = x² - 5x + 6

∴ α + β = - (-5)/1 = 5 and αβ = 6/1 = 6

Let S and P denote respectively the sum and product of the zeros of the polynomial whose zeros are 2α and 2β. Then,

S = 2α + 2β

= 2(α + β) = 2 × 5 = 10

P = 2α × 2β

= 4αβ = 4 × 6 = 24

Hence,

The required polynomial g(x) is given by

g(x) = k(x² - Sx + P) or, g(x) = k(x² - 10x + 24), where k is any non zero real number.