Question

The nature of roots of the equation
$(a+b+c)x^2-2(a+b)x+(a+b-c)=0(a,b,cϵQ)$

• A. Rational
• B. Irrational
• C. Imaginary
• D. None

Answer 1

The correct option is A Rational

Here:

$a=(a+b+c)$

$b=-2(a+b)$

$c=(a+b-c)$

$\therefore$ Determinant

$D=b^2-4ac$

$=[-2(a+b){]}^2-4[(a+b+c)(a+b-c)]$

$=\left[4\right(a+b{)}^2]-4[a^2+ab-ac+ab+b^2-bc+ac+bc-c^2]$

$=4(a^2+2ab+b^2)-4[a^2+ab-ac+ab+b^2-c^2]$

$=4a^2+8ab+4b^2-4a^2-4b^2+4c^2-8ab$

$=4c^2$

$\therefore D\ge 0$

$\therefore$ Roots of equation are rational.