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

Example 1- Find the value of λ for which the equation x2 + 2x + 3λ = 0 and 2x2 + 3x + 5λ = 0 may have a common root.

Solution-Let α be the common root of the given equations.
∴ α2 + 2α + 3λ = 0 ……….(i)
2 + 3α + 5λ = 0 ……….(ii)

Solving (i) and (ii) by cross-multiplication-
α2/(10λ – 9λ) = α/(6λ – 5λ) = 1/(3 – 4)
⇒ α2/λ = α/λ = 1/-1

Taking first two equations:
α2/λ = α/λ
⇒ α = 1

Taking last two equations:
α/λ = -1
⇒ 1/λ = -1 (∵ α = 1)
⇒ λ = -1
Example 2- If x2 – px + q = 0 and x2 – 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 x2 – ax + b = 0 are α, α.
∴ α + α = a and αα = b ⇒ α = a/2 and α2 = b.

Let α, β be the roots of equation x2 – 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 – α2 = q
⇒ p (a/2) – b = q
⇒ b + q = ap/2

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