Question

For$f\left(x\right)=a{x}^2+bx+c,a\ne 0,$ the conditions for which the graph is a downward opening parabola and $f\left(x\right)=0$ have a unique root with multiplicity $2$ is

• A. $a>0,D<0$
• B. $a<0,D<0$
• C. $a>0,D=0$
• D. $a<0,D=0$

Answer 1

The correct option is D $a<0,D=0$

Given: $f\left(x\right)=a{x}^2+bx+c,a\ne 0$
For graph to be a downward opening parabola we must have $a<0$
Also, the equation has one root with a multiplicity $2.$ This means the equation has identical real roots which occurs when discriminant $D=0.$