Question

If a, b, c are real numbers such that ac $\ne$ 0, then show that at least one of the equations $a{x}^2$ + bx + c = 0 and - $a{x}^2$ + bx + c = 0 has real roots.

Answer 1

Given $\{a,b,c\}ϵR$

and $ac\ne 0$

Let First eqn is $a{x}^2+bx+c=0$

and other is $-a{x}^2+bx+c=0$

and $D_1$ and $D_2$ are the discriminant of these eqn resp

Now, $=D_1=b^2-4ac$

$D_2=b^2+4ac$

case $i)ac<0$

then $D_1=b^2-4ac>0$

$D_2=b^2-4ac\to$ cant say

case $ii)ac>0$

then $D_1=b^2-4ac\to$ cant say

$D_2=b^2+4ac>0$

We can clearly observe that in both case atleast one equation have $D>0$ i.e. having real roots