Question

if x^2+ax+bc=0 and x^2+bx+ca=0 have a common root, then a+b+c

Answer 1

HELLO THERE!

Equation 1: x² + ax + bc

Equation 2: x² + bx + ac

These two have a common root.

Let this root be α.

Hence, α satisfies both equation 1 and 2.

Put α in Equation 1:

α² + aα + bc = 0

Put α in Equation 2:

α² + bα + ac = 0

Subtracting 2 from 1, we get:

α² + aα + bc - α² - bα - ac = 0

=> α(a - b) + c(b - a) = 0.

$\implies \alpha = \frac{-c(b-a)}{a-b} \\\\\implies \alpha = \frac{c(a-b)}{a-b} \\\\\implies \alpha = c$

Hence, the common root is c.

For equation 1, if the second root is β,

α + β = -b/a

=> α + β = -a

=> β = -a - c

=> a + c = -β ....(iii)

Also,

α β = bc

=> β = bc/c = b

Hence, from (iii),

a + c = -b

=> a + b + c = 0.

For equation 2, if the other root is γ,

α + γ = -b/a

=> α + γ = -b

=> γ = -b - c.

=> b + c = -γ .....(iv)

Also, α γ = ac

=> γ = ac/c = a

So, from (iv),

b + c = -a

=> a + b + c = 0

So, from both the equations, we get:

a + b + c = 0.

THANKS!

Answer 2

thanks for asking this question