Question

If the roots of the equation $a{x}^2+bx+c=0$ be $\frac{k+1}{k}$ and $\frac{k+2}{k+1}$, then $(a+b+c{)}^2=$

• A. $b^2-4ac$
• B. $c^2-4ab$
• C. $b^2-4ab$
• D. None

Answer 1

The correct option is A $b^2-4ac$

$\frac{k+1}{k}+\frac{k+2}{k+1}=\frac{-b}{a}$ ....(1)
And $\frac{k+2}{k}=\frac{c}{a}\Rightarrow k=\frac{2a}{c-a}$
$\frac{k+2}{k}=\frac{c}{a}\Rightarrow k=\frac{2a}{c-a}$ ....(2)
Substitute the value of $k$ from $\left(2\right)$ in $\left(1\right)$ $\frac{c+a}{2a}+\frac{2c}{c+a}=-\frac{b}{a}$
$\Rightarrow a(c+a{)}^2+4a^{2}c=-2abc-2a^{2}b$
$\Rightarrow (c+a{)}^2+4ac=-2bc-2ab$
$\Rightarrow (a+b+c{)}^2=b^2-4ac$