Quadratic equations

(x-a)(x-b) = c

x² -(a+b) + ab = c

x² -(a+b) + ab - c = 0

α + β = a + b and αβ = ab - c           ... (1)

Now, (x - α)(x - β) + c = a

x² -(α + β)x + αβ + c = a

Substituting from (1),

x² -(a + b)x + ab - c + c = a

x² -(a + b)x + ab - a = 0

The roots are,

= {(a+b)±[(a+b)² - 4a(b-1)]}/2

= {(a+b)±[(a-b)²+4a)]}/2

Regards,

Team,

TopperLearning.