We will discuss how to derive the conditions for common root or roots of quadratic equations that can be two or more.
Condition for one common root:
Let the two quadratic equations are a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0
Now we are going to find the condition that the above quadratic equations may have a common root.
Let α be the common root of the equations a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0. Then,
a1α^2 + b1α + c1 = 0
a2α^2 + b2α + c2 = 0
Now, solving the equations a1α^2 + b1α + c1 = 0, a2α^2 + b2α + c2 = 0 by cross-multiplication, we get
α^2/b1c2 - b2c1 = α/c1a2 - c2a1 = 1/a1b2 - a2b1
⇒ α = b1c2 - b2c1/c1a2 - c2a1, (From first two)
Or, α = c1a2 - c2a1/a1b2 - a2 b1, (From 2nd and 3rd)
⇒ b1c2 - b2c1/c1a2 - c2a1 = c1a2 - c2a1/a1b2 - a2b1
⇒ (c1a2 - c2a1)^2 = (b1c2 - b2c1)(a1b2 - a2b1), which is the required condition for one root to be common of two quadratic equations.
The common root is given by α = c1a2 - c2a1/a1b2 - a2b1 or, α = b1c2 - b2c1/c1q2 - c2a1
Note: (i) We can find the common root by making the same coefficient of x^2 of the given equations and then subtracting the two equations.
(ii) We can find the other root or roots by using the relations between roots and coefficients of the given equations
Condition for both roots common:
Let α, β be the common roots of the quadratic equations a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0. Then
α + β = -b1/a1, αβ = c1/a1 and α + β = -b2/a2, αβ = c2/a2
Therefore, -b/a1 = - b2/a2 and c1/a1 = c2/a2
⇒ a1/a2 = b1/b2 and a1/a2 = c1/c2
⇒ a1/a2 = b1/b2 = c1/c2
This is the required condition.
Solved examples to find the conditions for one common root or both common roots of quadratic equations:
1. If the equations x^2 + px + q = 0 and x^2 + px + q = 0 have a common root and p ≠ q, then prove that p + q + 1 = 0.
Let α be the common root of x^2 + px + q = 0 and x^2 + px + q = 0.
α^2 + pα + q = 0 and α^2 + pα + q = 0.
Subtracting second form the first,
α(p - q) + (q - p) = 0
⇒ α(p - q) - (p - q) = 0
⇒ (p - q)(α - 1) = 0
⇒ (α - 1) = 0, [p - q ≠0, since, p ≠ q]
⇒ α = 1
Therefore, from the equation α^2 + pα + q = 0 we get,
1^2 + p(1) + q = 0
⇒ 1 + p + q = 0
⇒ p + q + 1 = 0 Proved
2. Find the value(s) of λ so that the equations x^2 - λx - 21 = 0 and x^2 - 3λx + 35 = 0 may have one common root.
Let α be the common root of the given equations, then
α^2 - λα - 21 = 0 and α^2 - 3λα + 35 = 0.
Subtracting second form the first, we get
2λα - 56 = 0
2λα = 56
α = 56/2λ
α = 28/λ
Putting this value of α in α^2 - λα - 21 = 0, we get
(28/λ)^2 - λ * 28/λ - 21 = 0
(28/λ)^2 - 28 - 21 = 0
(28/λ)^2 - 49 = 0
16 - λ^2 = 0
λ^2 = 16
λ = 4, -4
Therefore, the required values of λ are 4, -4.