Question

If α and β are the zeros of the polynomial f(x)=x^2+x-2,find the value of (1/α-1/β).

Answer 1

$\large{\underline{\underline{\mathfrak{\green{\sf{Solution:-}}}}}}$.

• $\boxed{\red{\:(\frac{3}{2})}}$

$\large{\underline{\underline{\mathfrak{\green{\sf{Given\:Here:-}}}}}}$.

• $\:f(x)\:=\:x^2+x-2$

• $\alpha\: and \: \beta\:are\:Roots$

$\large{\underline{\underline{\mathfrak{\green{\sf{Find\:Here:-}}}}}}$.

• $\:Value\:of\:\frac{1}{\alpha}\:-\frac{1}{\beta}$

$\large{\underline{\underline{\mathfrak{\green{\sf{Explanation:-}}}}}}$.

We know that ,

$\large\red{\boxed{\:Sum\:Of\:Root\:=\frac{-(cofficient\:of\:x)}{(cofficient\:of\:x^2)}}}$

$\implies\:(\alpha\:+\beta)\:=\frac{-(1)}{1}$

$\implies\:(\alpha\:+\beta)\:=\:-1$.........(1)

Again,

$\large\red{\boxed{\:Product\:of\:Roots\:=\frac{Constant\:part}{Cofficient\:of\:x^2}}}$

$\implies\:(\alpha\beta)\:=\frac{-2}{1}$

$\implies\:(\alpha\beta)\:=\:-2$.........(2)

We Known,

• $\large\red{\boxed{\:(x-y)\:=\sqrt{(x+y)^2-4xy}}}$

So,

$\implies\:(\alpha\:-\beta)\:=\sqrt{(\alpha+\beta)^2-4\alpha\beta}$

Keep value by (1) and (2)

$\implies\:(\alpha\:-\beta)\:=\sqrt{(-1)^2-4(-2)}$

$\implies\:(\alpha\:-\beta)\:=\sqrt{(1+8)}$

$\implies\:(\alpha\:-\beta)\:=\sqrt{9}$

$\implies\boxed{\:(\alpha\:-\beta)\:=\:3}$.....(3)

We Find Here:-

$\large\red{\boxed{\frac{1}{\alpha}\:-\frac{1}{\beta}}}$

$\implies\:(\frac{\beta\:-\alpha}{\alpha\beta}$

keep value by (2) and (3) ,

$\implies\:\frac{-(3)}{-2}$

$\implies\boxed{\:\frac{3}{2}}$

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