Question

The condition that one root of the equation $a{x}^2+bx+c=0$ exceeds the other by $p$ is

• A. $a^2{p}^2=4ac$
• B. $a^2{p}^2=b^2+4ac$
• C. $a^2{p}^2=b^2-4ac$
• D. $ap=b^2+4ac$

Answer 1

The correct option is C $a^2{p}^2=b^2-4ac$

$Let\alpha ,\beta betherootsoftheequationa{x}^2+bx+c=0⟹\alpha +\beta =-\frac{b}{a}and\alpha \beta =\frac{c}{a.}Now|(\alpha -\beta )|=por\Rightarrow (\alpha -\beta {)}^2=p^2\Rightarrow (\alpha +\beta {)}^2-4\alpha \beta =p^2\Rightarrow {(-\frac{b}{a})}^2-4\frac{c}{a}=p^2\Rightarrow b^2-4ac=a^2{p}^2\Rightarrow a^2{p}^2=b^2-4acAnswer-OptionC$