**Condition for Common Roots – Problems**

**Condition for Common Roots:** Let ax² + bx + c = 0 and px² + qx +r
= 0 have a common roots α

Then ax² + bx + c = 0 and px² + qx + r = 0

Solving equations

\(\frac{{{\alpha }^{2}}}{br-cq}=\frac{\alpha }{pc-ar}=\frac{1}{aq-bp}\).

Condition for both roots to be common is

a/p = b/q = c/r

**Example1:** If at least one root of the equation
x³ + ax² + bx + c = 0 remains unchanged, when a, b and c are decreased by one,
then which one is always a root of the given equation?

**Solution: **x³ + ax² + bx + c = 0

and x³ + (a – 1)x² + (b-1)x + (c – 1) = 0have at least one common root, let common root be α

α³ + aα² + bα + c = 0

and α³ + aα² + bα + c – α² – α –- 1 = 0

⇒ α² + α + 1 = 0

⇒ α = ω, ω² (where, ω and ω² are the cube roots of unity)

**Example2:** If the equation 2x² + 3x + 5k = 0 and
x² + 2x + 3 k = 0 have a common root, then k is equal to?

**Solution:**

Given equations are 2x² + 3x + 5k = 0 and x² + 2x + 3k = 0 have a common root, if

\(\frac{{{x}^{2}}}{(9-10)k}=\frac{x}{(5-6)k}=\frac{1}{(4-3)}\).

\(\frac{{{x}^{2}}}{-k}=\frac{x}{-k}=1\).

x²/ – k = 1

x/ – k = 1

x² = – k

x = – k

(or)

K = 0, 1

2(– k) + 3(- k) + 5k = 0

-2k – 3k + 5k = 0

**Example3:** If each pair of the equation x² + ax
+ b = 0, x² + bx + c = 0 has a common roots, then product of all common roots
is ?

**Solution:**

Let the roots be α, β, β, γ and γ, α then

αβ = b

βγ = c

and γα = a

(αβγ)² = abc

αβγ = √(abc)