Question

If
$\alpha$
and
$\beta$
are zeroes of
$5x {}^{2} + 5x + 1$
then find
$\alpha { }^{ - 1} + \beta {}^{ - 1}$
​

Answer 1

$★ {\pmb{\underline{\sf{Required \ Solution ... }}}}$

As We know that $\alpha$ and $\beta$ are zeroes of ${\tt{ 5x^2 + 5x + 1 }}$.

»Firstly, We've to Find the sum of the Zeroes of the Quadratic Polynomial as:

$\colon\implies{\tt{ \alpha + \beta = \dfrac{-b}{a} }} \\ \\ \colon\implies{\tt{ \cancel{ \dfrac{-5}{5} } = -1 }}$

»Now, We should find the Product of the Zeroes of the Quadratic Polynomial as:

$\colon\implies{\tt{ \alpha \beta = \dfrac{c}{a} }} \\ \\ \colon\implies{\tt{ \dfrac{1}{5} }}$

Now Finally, It's Time to get the value of the Desired Equation as well.

$\colon\implies{\tt{ \dfrac{1}{ \alpha } + \dfrac{1}{ \beta } }} \\ \\ \colon\implies{\tt{ \dfrac{ \alpha + \beta}{ \alpha \beta } }} \\ \\ \colon\implies{\tt{ \dfrac{-1}{ \dfrac{1}{5} } }} \\ \\ \colon\implies{\tt{ \dfrac{-5}{1} = -5 }}$

Hence,

The value of $\alpha { }^{ - 1} + \beta {}^{ - 1}$ is ${\tt{ -5}}$.