## Quadratic Equations Common Roots:

Let the two quadratic equations a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0 have a common root α. Then, α will satisfy both the equations

∴ a_{1}α^{2} + b_{1}α + c_{1} = 0 and a_{2}α^{2} + b_{2}α + c_{2} = 0

By the method of cross-multiplication, we get

α^{2}/(b_{1}c_{2} – b_{2}c_{1}) = α/(c_{1}a_{2} – a_{1}c_{2}) = 1/(a_{1}b_{2} – a_{2}b_{1})

Now, the first two ratios give α = (b_{1}c_{2} – b_{2}c_{1})/(c_{1}a_{2} – a_{1}c_{2}) ……….(i)

The second and the third ratios give α = (c_{1}a_{2} – a_{1}c_{2})/(a_{1}b_{2} – a_{2}b_{1}) ……….(ii)

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

Now, from (i) and (ii), we get (b_{1}c_{2} – b_{2}c_{1})/(c_{1}a_{2} – a_{1}c_{2}) = (c_{1}a_{2} – a_{1}c_{2})/(a_{1}b_{2} – a_{2}b_{1})

⇒ (b_{1}c_{2} – b_{2}c_{1}) (a_{1}b_{2} – a_{2}b_{1}) = (c_{1}a_{2} – a_{1}c_{2})^{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 a_{1}x^{2} + b_{1}x + c_{1} = 0, then αβ = c_{1}/a_{1}

∴ β = (c_{1}/a_{1}) (1/α) = c_{1} (c_{1}a_{2} – a_{1}c_{2})/a_{1} (b_{1}c_{2} – b_{2}c_{1}) ……….(iii)

Again, if γ be the other root of the quadratic equation a_{2}x^{2} + b_{2}x + c_{2} = 0, then αγ = c_{2}/a_{2}

∴ γ = (c_{2}/a_{2}) (1/α) = c_{2} (c_{1}a_{2} – a_{1}c_{2})/a_{2} (b_{1}c_{2} – b_{2}c_{1}) ……….(iv)

If the quadratic equations a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0 have both the roots common, then α + β = α + γ

or, -b_{1}/a_{1} = -b_{2}/a_{2} ⇒ a_{1}/a_{2} = b_{1}/b_{2} ……….(v)

and also αβ = αγ ⇒ c_{1}/a_{1} = c_{2}/a_{2} ⇒ a_{1}/a_{2} = c_{1}/c_{2} ……….(vi)

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

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

Example 1- Find the value of λ for which the equation x^{2} + 2x + 3λ = 0 and 2x^{2} + 3x + 5λ = 0 may have a common root.Solution-Let α be the common root of the given equations.∴ α ^{2} + 2α + 3λ = 0 ……….(i)2α ^{2} + 3α + 5λ = 0 ……….(ii)Solving (i) and (ii) by cross-multiplication- α ^{2}/(10λ – 9λ) = α/(6λ – 5λ) = 1/(3 – 4)⇒ α ^{2}/λ = α/λ = 1/-1Taking first two equations: α ^{2}/λ = α/λ⇒ α = 1 Taking last two equations: α/λ = -1 ⇒ 1/λ = -1 (∵ α = 1) ⇒ λ = -1 |

Example 2- If x^{2} – px + q = 0 and x^{2} – ax + b = 0 have one common root and the second equation has equal toots then prove that b + q = ap/2.Solution- Let α be the common root of both the equations.Also, the roots of equation x ^{2} – ax + b = 0 are α, α.∴ α + α = a and αα = b ⇒ α = a/2 and α ^{2} = b.Let α, β be the roots of equation x ^{2} – px + q = 0.∴ α + β = p and αβ = q ⇒ β = p – α ………….(i) and β = q/α ………….(ii) From (i) and (ii)- p – α = q/α ⇒ pα – α ^{2} = q⇒ p (a/2) – b = q ⇒ b + q = ap/2 |