Question

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

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

Answer 1

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

For quadratic equation $a{x}^2+bx+c=0,a\ne 1$
Nature of roots is given by discriminant $D=b^2-4c$ 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 it not being a perfect square of a rational number
$\Rightarrow$ Roots are irrational and unequal.
$\left(iv\right)D>0$ and it being a perfect square of a rational number
$\Rightarrow$ Roots are rational and unequal
$\left(v\right)D>0\&D$ is a perfect square and $a=1;b,c\in Z$
$\Rightarrow$ Real, unequal and integral roots

Statement 1 & 2: If $D=0$ roots are real and equal.
Here, $b\in Q\Rightarrow$
Thus, the root will be $-b\in Q$
Hence, $D=0\Rightarrow$ Roots are real and equal.

Statement 3: If $D>0,\Rightarrow$ Roots are integers and unequal which is not always true since $b,c$ can be any rational numbers.

Statement 4: If $D>0,$ roots are rational and unequal which is not always true as it all depends on whether $D$ is perfect square of a rational number or not.