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) 2α2 + 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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