Question

For the quadratic equation $a{x}^2+bx+c=0;a,b,c\in R,$
which of the following is/are true ? (where $\Delta =b^2-4ac$,)

• A. roots are negative, if all three of $a,b,c$ have the same sign and $\Delta >0$
• B. roots are negative, if $a,c$ have the same sign but $b$ is of opposite sign and $\Delta >0$
• C. if $c=0,$ then atleast one root is zero.
• D. $x=0$ is a repeated root, if $b=c=0$

Answer 1

Answer: (A), (C), (D)

$x=0$ is a repeated root, if $b=c=0$

$a{x}^2+bx+c=0;a,b,c\in R$
If $b=c=0$
$\Rightarrow a{x}^2=0$
$\Rightarrow x=0,0$

If $c=0,$ then
$a{x}^2+bx=0$
$\Rightarrow (ax+b)x=0$
$\Rightarrow x=0,-\frac{b}{a}$

For real roots, $\Delta =b^2-4ac\ge 0$
Let roots are $m$ and $n$
$m+n=-\frac{b}{a},mn=\frac{c}{a}$
Since, roots are negative,
Therefore, both $\frac{b}{a}$ and $\frac{c}{a}$ are positive.
$\frac{b}{a}\to +ve\Rightarrow a,b\to +ve\text{or}a,b\to -ve$
$\frac{c}{a}\to +ve\Rightarrow a,c\to +ve\text{or}a,c\to -ve$
$\Rightarrow$ Either $a,b,c\to +ve\text{or}a,b,c\to -ve$

Hence, if roots are negative, then all three of $a,b,c$ have the same sign.