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.

**Solution:**

Let α be the common root of x^2 + px + q = 0 and x^2 + px + q = 0.

Then,

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.

**Solution:**

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.

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