Question

if the equations x²+2x+3 =0 and ax²+bx+c=0 , a, b, c ∈ R, have a common root, then a:b:c is
a) 1:2:3 b) 3:2:1
c) 1:3:2 d) 3:1:2​

Answer 1

Solution:

$\sf{x^{2}+2x+3=0}$

$\implies$ $\sf{d=({2})^{2}-4\times3}$

$\implies$ $\sf{d= 4 - 12}$

$\implies$ $\sf{d = - 8}$

Where: d < 0

Note: Roots are imaginary and imaginary roots always exists in pairs (conjugate).

Therefore:

$\sf{ax^{2}+bx+c=0}$ and $\sf{x^{2}+2x+3=0}$ are identical.

So:

$\boxed{\sf{\frac{a}{1}=\frac{b}{2}=\frac{c}{3}}}$

Hence:

$\implies$ $\boxed{\sf{a : b : c = 1 : 2 : 3}}$

Therefore:

Correct option: a) 1 : 2 : 3

Answer 2

Answer:

1 : 2 : 3

Option a . is correct .

Step-by-step explanation:

Given :

x² + 2 x + 3 = 0

First finding nature of roots .

Applying direct formula for find x .

x = - b ± √  ( b² - 4 ac )  / 2 a

x = - 2 ± √  ( 2² - 4 × 1 × 3 )  / 2 × 1

x = - 2 ± √  ( 4 - 12 )  / 2 × 1

x = - 2 ± √  ( - 8  )  / 2

x = - 2 ±  2 i √  ( 2  )  / 2

x = 2 ( - 1 ±  i √2 )   / 2

x =  ( - 1 ±  i √2 )  

Since root of x² + 2 x + 3 = 0  is imaginary .

So common root of a x² + bx + c = 0 will be same as x² + 2 x + 3 = 0 .

i.e.  a : b : c = 1 : 2 : 3

Thus we get answer .