Let the two quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have a common root α. Then, α will satisfy both the equations

∴ a1α2 + b1α + c1 = 0 and a2α2 + b2α + c2 = 0

By the method of cross-multiplication, we get

α2/(b1c2 – b2c1) = α/(c1a2 – a1c2) = 1/(a1b2 – a2b1)

Now, the first two ratios give α = (b1c2 – b2c1)/(c1a2 – a1c2) ……….(i)

The second and the third ratios give α = (c1a2 – a1c2)/(a1b2 – a2b1) ……….(ii)

The values of the common roots are given by (i) and (ii)-

Now, from (i) and (ii), we get (b1c2 – b2c1)/(c1a2 – a1c2) = (c1a2 – a1c2)/(a1b2 – a2b1)

⇒ (b1c2 – b2c1) (a1b2 – a2b1) = (c1a2 – a1c2)2, which gives the condition for the two quadratic equations to have a common root.

Now, if β be the other root of the quadratic equation a1x2 + b1x + c1 = 0, then αβ = c1/a1

∴ β = (c1/a1) (1/α) = c1 (c1a2 – a1c2)/a1 (b1c2 – b2c1) ……….(iii)

Again, if γ be the other root of the quadratic equation a2x2 + b2x + c2 = 0, then αγ = c2/a2

∴ γ = (c2/a2) (1/α) = c2 (c1a2 – a1c2)/a2 (b1c2 – b2c1) ……….(iv)

If the quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have both the roots common, then α + β = α + γ

or, -b1/a1 = -b2/a2 ⇒ a1/a2 = b1/b2 ……….(v)

and also αβ = αγ ⇒ c1/a1 = c2/a2 ⇒ a1/a2 = c1/c2 ……….(vi)

From (v) and (vi) we get the condition for the two quadratic equations to have both the roots common as-

a1/a2 = b1/b2 = c1/c2