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. b is greater than zero

Answer 1

Answer: (B), (D)

The correct options are
B

Larger root is – 1

D

b is greater than zero

Given $\alpha >\beta ;0<a<c\Rightarrow a>0,c>0andc>aanda-b+c=0\Rightarrow f(-1)=0\Rightarrow x=-1$ 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 = $\frac{-b}{a}<0\Rightarrow b>0.$