Question

Let a,b,ceR and a *0. Then the roots of ar? +bx+c=0 are non-real complex
numbers if and only if ax? + bx+c and a have the same sign for all xeR.​

Answer 1

Answer:

Roots of the quadratic equation ax

2

+bx+c=0 are non real.

⇒b

2

−4ac<0 -----(1)

Now, for the expression a

2

x

2

+abx+ac,

Δ=a

2

b

2

−4a

3

c=a

2

(b

2

−4ac)<0 from (1)

Since, discriminant value is less than zero, coefficient of x

2

and expression always have same sign.

∴a

2

x

2

+abx+ac>0