Question

find the roots of x^2 - 2 x - (r^2 - 1) = 0​

Answer 1

x² - 2x - (r² - 1) = 0

or, x² - 2x - r² + 1 = 0

or, (x² - 2x + 1) - r² = 0

or, (x - 1)² - r² = 0

use formula, a² - b² = (a - b)(a + b)

so, (x - 1)² - r² = 0

or, (x - 1 - r)(x - 1 + r) = 0

or, x = 1 + r, 1 - r

hence, root of x² - 2x - (r² - 1) = 0 is 1 - r and 1 + r .

Answer 2

Given Quadratic equation is,

x² - 2x - (r² - 1) = 0

Let the roots of the equation be α and β.

α + β = -(-2)/1 = 2

αβ = c/a = -(r² - 1)/1 = (1 - r²)

(α + β)² - (α - β)² = 4αβ

(2)² - (α - β)² = 4 × (1 - r²)

4 - 4(1 - r²) =  (α - β)²

(α - β)² = 4r²

(α - β) = ± 2r

When  (α - β) = 2r, then, 2α = 2 + 2r

α = 1 + r , and β = 1 - r

When (α - β) = -2r, then, 2α = 2 - 2r

α = 1 - r and β = 1 + r

Hope it helps.