Question

Find a for which one root is positive and other root is negative for $-x^2-(3a-2)x+a^2+1=0$

• A. $\varphi$
• B. $R-0$
• C. None of the above
• D. $R$

Answer 1

The correct option is D $R$

Step-1: Making leading coefficient positive.
Let $f\left(x\right)=x^2+(3a-2)x-(a^2+1)$ then,
Step-2:
SInce one root is negative and ohter is poisitve, it means one root is less than 0 and other is greater than 0.
Uaing, $a.f\left(d\right)<0$
$$\begin{array}{c}\text{we get}f\left(0\right)<0\\ \Rightarrow -(a^2+1)<0\\ \text{which is true for all real values of a.}\\ \Rightarrow a\in R\end{array}$$