Question

Condition for 2 quadratic equations to have a common root

Answer 1

Let α, β be the common roots of the quadratic equations a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0. Then

α + β = -b1/a1, αβ = c1/a1 and α + β = -b2/a2, αβ = c2/a2

Therefore, -b/a1 = - b2/a2 and c1/a1 = c2/a2

⇒ a1/a2 = b1/b2 and a1/a2 = c1/c2

⇒ a1/a2 = b1/b2 = c1/c2

This is the required condition.

Answer 2

Hope u understand by the help of pictures