Quadratic Equations Common Roots

Quadratic Equations Common Roots:

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