if alpha and beta are the roots of x^2 + bx + c =0, then the value of Alpha^2/ beta^2 +beta^2 /Alpha^2 is....
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if alpha and beta are the roots of x^2 + bx + c =0, then the value of Alpha^2/ beta^2 +beta^2 /Alpha^2 is....
Please solve it... it's urgent
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[tex]\huge\red{\underline{\underline{\pink{Ans}\red{wer:}}}}[/tex]
[tex]\sf{The \ value \ of \ \frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}} \ is \ \frac{b^{4}-4b^{2}c+2c^{2}}{c^{2}}}[/tex]
[tex]\sf\orange{}[/tex]
[tex]\sf{The \ given \ quadratic \ equation \ is}[/tex]
[tex]\sf{\implies{x^{2}+bx+c=0}}[/tex]
[tex]\sf{\implies{Roots \ are \ \alpha \ and \ \beta}}[/tex]
[tex]\sf\pink{To \ find:}[/tex]
[tex]\sf{\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}}[/tex]
[tex]\sf\green{\underline{\underline{Solution:}}}[/tex]
[tex]\sf{The \ given \ quadratic \ equation \ is}[/tex]
[tex]\sf{\implies{x^{2}+bx+c=0}}[/tex]
[tex]\sf{Here, \ a=1, \ b=b \ and \ c=c}[/tex]
[tex]\sf{Sum \ of \ roots=\frac{-b}{a}}[/tex]
[tex]\sf{\implies{\therefore{\alpha+\beta=-b...(1)}}}[/tex]
[tex]\sf{Product \ of \ roots=\frac{c}{a}}[/tex]
[tex]\sf{\implies{\therefore{\alpha×\beta=c...(2)}}}[/tex]
[tex]\sf{\implies{\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}}}[/tex]
[tex]\sf{\implies{\frac{\alpha^{4}}{\alpha^{2}×\beta^{2}}+\frac{\beta^{4}}{\alpha^{2}×\beta^{2}}}}[/tex]
[tex]\sf{\implies{\frac{\alpha^{4}+\beta^{4}}{(\alpha×\beta)^{2}}}}[/tex]
[tex]\sf{By \ identity}[/tex]
[tex]\sf{a^{4}+b^{4}=(a+b)^{4}-4ab(a^{2}+b^{2})-6(ab)^{2}}[/tex]
[tex]\sf{\therefore{a^{4}+b^{4}=(a+b)^{4}-4ab[(a+b)^{2}-2ab]-6(ab)^{2}}}[/tex]
[tex]\sf{\implies{\frac{(-b)^{4}-4(c)[(-b)^{2}-2c]-6c^{2}}{c^{2}}}}[/tex]
[tex]\sf{\implies{\frac{b^{4}-4b^{2}c+8c^{2}-6c^{2}}{c^{2}}}}[/tex]
[tex]\sf{\implies{\frac{b^{4}-4b^{2}c+2c^{2}}{c^{2}}}}[/tex]
[tex]\sf\purple{\tt{\therefore{The \ value \ of \ \frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}} \ is \ \frac{b^{4}-4b^{2}c+2c^{2}}{c^{2}}}}}[/tex]