Question

if alpha and beta are roots of ax² + bx+c =0 then 1/alpha²+ 1/beta² =​

Answer 1

$\bf{\Huge{\boxed{\rm{\pink{ANSWER\::}}}}}$

$\bf{\Large{\underline{\bf{Given\::}}}}$

If α & β are roots of ax² + bx + c = 0.

$\bf{\Large{\underline{\bf{To\:\:find\::}}}}$

$\bf{\Large{\sf{\frac{1 }{\alpha^{2} } +\frac{1}{\beta ^{2} } ? }}}$

$\bf{\Large{\underline{\tt{\red{Explanation\::}}}}}$

We have, ax² + bx + c = 0

$\bf{\Large{\boxed{\sf{\green{Sum\:of\:roots\::}}}}}$

$\mapsto\rm{\alpha +\beta \:=\:\frac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} } }$

$\mapsto\rm{\alpha +\beta \:=\:-\frac{b}{a} }$

$\bf{\Large{\boxed{\sf{\green{Product\:of\:roots\::}}}}}$

$\mapsto\rm{\alpha \beta \:=\:\frac{Constant\:term}{Coefficient\:of\:x^{2} } }$

$\mapsto\rm{\alpha \beta \:=\:\frac{c}{a} }$

$\leadsto\sf{Quadratic\:polynomial\:g(x)=\:x^{2} -(\alpha +\beta )^{2} +\alpha \beta }$

Now,

$\longmapsto\tt{\frac{1}{\alpha ^{2} } +\frac{1}{\beta ^{2} } =x^{2} -(\frac{1}{\alpha ^{2} } +\frac{1}{\beta ^{2} } )x+\frac{1}{\alpha ^{2} } \:\frac{1}{\beta ^{2} } =0}$

$\longmapsto\tt{x^{2} -(\frac{\alpha ^{2} +\beta ^{2} }{\alpha ^{2} \beta ^{2} } )x+\frac{1}{(\alpha\beta )^{2} } =0}$

$\longmapsto\tt{x^{2} -[\frac{(\alpha+\beta)^{2} -2\alpha \beta }{(\alpha \beta )^{2} } ]x+\frac{1}{(\alpha \beta)^{2} } =0}$

$\longmapsto\tt{x^{2} -(\frac{\frac{b^{2} }{\cancel{a^{2}} } -2*\frac{c}{a} }{\frac{c^{2} }{\cancel{a^{2}} } } )x+\frac{a^{2} }{c^{2} } =0}$

$\longmapsto\tt{x^{2} -(\frac{b^{2} -2ac}{c^{2} } )x+\frac{a^{2} }{c^{2} } =0}$

$\longmapsto\sf{c^{2} x^{2} -(b^{2} -2ac)x\:+a^{2} =0}$

Thus,

$\bf{\Large{\boxed{\tt{c^{2} x^{2} -(b^{2} -2ac)x\:+a^{2} =0}}}}}$

Answer 2

If α & β are roots of ax² + bx + c = 0

$\tt\green{To\:\:find\::}$

$\tt{\frac{1 }{\alpha^{2} } +\frac{1}{\beta ^{2} } }$

$\tt\green{Explanation\::}$

We have, ax² + bx + c = 0

So the Sum of roots is

${\alpha +\beta \:=\:\frac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} } }$

${\alpha +\beta \:=\:-\frac{b}{a} }$

$\bf{\Large{\boxed{\tt{\blue{Product\:of\:roots\::}}}}}$

${\alpha \beta \:=\:\frac{Constant\:term}{Coefficient\:of\:x^{2} } }$

${\alpha \beta \:=\:\frac{c}{a} }$

$\tt{Quadratic\:polynomial\:g(x)=\:x^{2} -(\alpha +\beta )^{2} +\alpha \beta }$

__________________________

$\tt{\frac{1}{\alpha ^{2} } +\frac{1}{\beta ^{2} } =x^{2} -(\frac{1}{\alpha ^{2} } +\frac{1}{\beta ^{2} } )x+\frac{1}{\alpha ^{2} } \:\frac{1}{\beta ^{2} } =0}$

$\tt{x^{2} -[\frac{(\alpha+\beta)^{2} -2\alpha \beta }{(\alpha \beta )^{2} } ]x+\frac{1}{(\alpha \beta)^{2} } =0}$

$\tt{x^{2} -(\frac{\frac{b^{2} }{\cancel{a^{2}} } -2*\frac{c}{a} }{\frac{c^{2} }{\cancel{a^{2}} } } )x+\frac{a^{2} }{c^{2} } =0}$

$\tt{x^{2} -(\frac{b^{2} -2ac}{c^{2} } )x+\frac{a^{2} }{c^{2} } =0}$

$\tt\green{c^{2} x^{2} -(b^{2} -2ac)x\:+a^{2} =0}$