Question

If x^2+x+1=0 and ax^2+bx+c=0 have a common root, prove that a=b=c

Answer 1

Step-by-step explanation:

If equation ax

2

+bx+c=0 & cx

2

+bx+a=0 have a common root prove a+b+c=0 or a+b+c=0

→ Let assume α is common root

put in equation (i) & (ii)

aα+bα+c=0 & cα

2

+bα+a=0

∴ Compare both the equation

aα

2

+bα+c=cα

2

+bα+a

∴aα

2

−cα

2

=a−c Substitute the value of α in equation

∴α

2

(a−c)=a−c

∴α

2

=1 ∴ we get, a+b+c=0(α+1) & a−b+c=0(for α=−2)

∴α=+1,−1

Answer 2

Answer:

For equation x

2

+2x+3=0

Δ=2

2

−4(1)(3)=4−12=−8<0 so both roots are imaginary .

Hence, the roots are non-real. They will exist in complex conjugate pairs.

As one of the roots is common to ax

2

+bx+c=0 , the other root will also be the complex conjugate of it.

Hence, the roots of the two equations will be the same.

Since a, b, c∈R.

If one root is common, then both roots are common .

Hence, 1a

= 2b

= 3c

a:b:c=1:2:3

Step-by-step explanation:

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