Question

If α and β are the zeros of the quadratic polynomial f(x) = x² – p(x + 1) – c, show that (α + 1)(β + 1) = 1 – c.

Answer 1

Since α and β are the zeros of the quadratic polynomial f(x) = x² – p (x + 1) – c

Then

x² - p(x + 1) - c

x² - px - p - c

`alpha+beta="-coefficient of x"/("coefficient of "x^2)`

`=(-(-p))/1`

= p

`alphabeta="constant term"/("coefficient of "x^2)`

`=(-p-c)/1`

= -p-c

We have to prove that (α + 1)(β +1) = 1 − c

(α + 1)(β +1) = 1 - c

(α + 1)β + (α +1)(1) = 1 - c

αβ + β + α + 1 = 1 - c

αβ + (α + β) + 1 = 1 - c

Substituting α + β = p and αβ = -p-c we get,

-p - c + p + 1 = 1 - c

1 - c = 1 - c

Hence, it is shown that (α + 1)(β +1) = 1 - c