Question

If the roots of the equation $a{x}^2+2bx+c=0$ imaginary, then the roots of the equation $a{x}^2+2(a+b)x+(a+2b+c)=0$ will be:

• A. Imaginary
• B. Real
• C. Rational
• D. Irrational

Answer 1

The correct option is A Imaginary

$a{x}^2+bx+c=0$

For imaginary roots, $D<0$

$\Rightarrow (2b{)}^2-4ac<0$

$\Rightarrow b^2-ac<0$-Equation 1.

Since, $a{x}^2+2(a+b)x+(a2b+c)$

$D\equiv \left(2\right(a+b{)}^2)-4(a+2b+c\left)\right(a)$

$\equiv 4(a^2+b^2+2ab)-4(a^2+2ab+ac)$

$\equiv 4(b^2-ac)<0$

Here $D<0\Rightarrow$ imaginary roots.