Question

proof of condition of common roots

Answer 1

Let the two quadratic equations are a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0

Now we are going to find the condition that the above quadratic equations may have a common root.

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

a1α^2 + b1α + c1 = 0

a2α^2 + b2α + c2 = 0

Now, solving the equations a1α^2 + b1α + c1 = 0, a2α^2 + b2α + c2 = 0 by cross-multiplication, we get

α^2/b1c2 - b2c1 = α/c1a2 - c2a1 = 1/a1b2 - a2b1

⇒ α = b1c2 - b2c1/c1a2 - c2a1, (From first two)

Or, α = c1a2 - c2a1/a1b2 - a2 b1, (From 2nd and 3rd)

⇒ b1c2 - b2c1/c1a2 - c2a1 = c1a2 - c2a1/a1b2 - a2b1

⇒ (c1a2 - c2a1)^2 = (b1c2 - b2c1)(a1b2 - a2b1), which is the required condition for one root to be common of two quadratic equations.

The common root is given by α = c1a2 - c2a1/a1b2 - a2b1 or, α = b1c2 - b2c1/c1q2 - c2a1