Question

Let $\alpha$ and $\beta (\alpha >\beta )$ be roots of the quadratic equation $a{x}^2+bx+c=0$.
If $0<a<c$ and $a+c=b$, then which of the following statement(s) is/are correct?

• A. Larger root is 1 .
• B. Larger root is -1 .
• C. Smaller root is $\frac{c}{a}$.
• D. bis greater than zero.

Answer 1

Answer: (B)

$\text{ Given }\alpha >\beta ;0<a<c\Rightarrow a>0,c>0\text{ and }c>a$
$\text{ an }a-b+c=0\Rightarrow f(-1)=0\Rightarrow x=-1\text{ is one root. }$

and product of root $=\frac{c}{a}>1$
Hence the other root must also be less than -1
$\Rightarrow$ larger root is -1 .

Also, $(-1)(\alpha )=\frac{c}{a}\Rightarrow$ smaller root $=\frac{-c}{a}$.
As sum of roots $=\frac{-b}{a}<0\Rightarrow b>0$.