Math, asked by viki37, 3 months ago

do it kal exam hai....​

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Answers

Answered by BrainlyEmpire
72

Given :-

\sf\ \alpha = \dfrac{-b+\sqrt{b^2-4ac}}{2a}\\ \\ \\ \sf\ \beta = \dfrac{-b-\sqrt{b^2-4ac}}{2a}

To prove :-

\sf\ \alpha+ \beta= \dfrac{-b}{a}\\ \\ \\ \sf\ \alpha \beta= \dfrac{c}{a}

Proof :-

\implies\sf\ \alpha+ \beta= \Bigg\{\dfrac{-b+\sqrt{b^2-4ac}}{2a}\Bigg\}+\Bigg\{\dfrac{-b-\sqrt{b^2-4ac}}{2a}\Bigg\}\\ \\ \\ \\ \implies\sf\ \alpha+\beta= \dfrac{-b+\cancel{\sqrt{b^2-4ac}}+(-b)-\cancel{\sqrt{b^2-4ac}}}{2a}\\ \\ \\ \\ \implies\sf\ \alpha+\beta= \cancel{\dfrac{-2b}{2a}}\\ \\ \\ \\ \implies\boxed{\purple{\sf \alpha+\beta= \dfrac{-b}{a}}}\ \ \ \ \sf( Hence\ Proved !!)

Now,

\implies\sf\ \alpha \beta= \Bigg\{\dfrac{-b+\sqrt{b^2-4ac}}{2a}\Bigg\}\Bigg\{\dfrac{-b-\sqrt{b^2-4ac}}{2a}\Bigg\}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \dfrac{(-b)^2-(\sqrt{b^2-4ac})^2}{2a\times 2a}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \dfrac{b^2- (b^2-4ac)}{4a^2}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \dfrac{\cancel{b^2}-\cancel{b^2}+4ac}{4a^2}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \cancel{\dfrac{4ac}{4a^2}}\\ \\ \\ \implies\boxed{\red{\sf \alpha \beta= \dfrac{c}{a}}}\ \ \ \ \sf (Hence\ Proved !!)

Answered by Anonymous
23

Answer:

Given :-

\sf\ \alpha = \dfrac{-b+\sqrt{b^2-4ac}}{2a}\\ \\ \\ \sf\ \beta = \dfrac{-b-\sqrt{b^2-4ac}}{2a}

To prove :-

\sf\ \alpha+ \beta= \dfrac{-b}{a}\\ \\ \\ \sf\ \alpha \beta= \dfrac{c}{a}

Proof :-

\implies\sf\ \alpha+ \beta= \Bigg\{\dfrac{-b+\sqrt{b^2-4ac}}{2a}\Bigg\}+\Bigg\{\dfrac{-b-\sqrt{b^2-4ac}}{2a}\Bigg\}\\ \\ \\ \\ \implies\sf\ \alpha+\beta= \dfrac{-b+\cancel{\sqrt{b^2-4ac}}+(-b)-\cancel{\sqrt{b^2-4ac}}}{2a}\\ \\ \\ \\ \implies\sf\ \alpha+\beta= \cancel{\dfrac{-2b}{2a}}\\ \\ \\ \\ \implies\boxed{\purple{\sf \alpha+\beta= \dfrac{-b}{a}}}\ \ \ \ \sf( Hence\ Proved !!)

Now,

\implies\sf\ \alpha \beta= \Bigg\{\dfrac{-b+\sqrt{b^2-4ac}}{2a}\Bigg\}\Bigg\{\dfrac{-b-\sqrt{b^2-4ac}}{2a}\Bigg\}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \dfrac{(-b)^2-(\sqrt{b^2-4ac})^2}{2a\times 2a}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \dfrac{b^2- (b^2-4ac)}{4a^2}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \dfrac{\cancel{b^2}-\cancel{b^2}+4ac}{4a^2}\\ \\ \\ \\ \implies\sf\ \alpha \beta= \cancel{\dfrac{4ac}{4a^2}}\\ \\ \\ \implies\boxed{\red{\sf \alpha \beta= \dfrac{c}{a}}}\ \ \ \ \sf (Hence\ Proved !!)

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