Question

Root of quadratic equation $a{x}^2+bx+c=0,a\ne 0$ and $a=1,b,c\in Q$.
Then which of the following is/are always true

• A. If $D<0$, roots are real and distinct.
• B. If $D>0$, roots are rational and unequal.
• C. If $D=0$, roots are real and equal.
• D. If $D>0$, roots are integer and unequal

Answer 1

The correct option is C If $D=0$, roots are real and equal.

For quadratic equation $a{x}^2+bx+c=0,a\ne 0$ Nature of roots is given by discriminant $D=b^2-4ac$ as $\left(i\right)D<0\Rightarrow \text{Non real or imaginary roots}\left(ii\right)D=0\Rightarrow \text{Real and equal roots}$ $\left(iii\right)D>0$ and D is not a perfect square of a rational number
$\Rightarrow$ roots are irrational and unequal.
$\left(iv\right)D>0$ and D is perfect square of a rational number
$\Rightarrow$ roots are rational and unequal
$\left(v\right)D>0,D$ is a perfect square and $a=1,b,c\in Z$
$\Rightarrow$Non real or imaginary roots}


Statement 1: If $D<0$, roots are real and distinct.
False because roots are imaginary for $D<0$ from $\left(i\right)$

Statement 2: If $D=0$, roots are real and equal.
True from $\left(ii\right)$


Statement 3: If $D>0$, roots are integer and unequal
not always true for b,c as any rational numbers.

Statement 4: If $D>0$, roots are rational and unequal.
not always true. Depends whether D is perfect square of a rational number or not.