Question

The nature of the roots of $\frac{1}{2}{x}^2+(a+b+c)x+ab+bc+ca=0$ are

• A. Real and Distinct
• B. Real and Equal
• C. Imginary and Complex
• D. Cant be determined

Answer 1

The correct option is A Real and Distinct

Roots of $\frac{1}{2}{x}^2+(a+b+c)x+ab+bc+ca=0$ are

Here $a=\frac{1}{2},b=a+b+c,c=ab+bc+ca$

For Nature of Roots

$b^2-4ac$

$⟹(a+b+c{)}^2-4\left(\frac{1}{2}\right)(ab+bc+ca)$

$⟹a^2+b^2+c^2+2ab+2bc+2ca-2ab-2bc-2ca$

$⟹a^2+b^2+c^2>0$

So the Roots are Real and Distinct