Question

if alpha and beta are the zeros of x^2+5x+6 find the value of alpha power -1 + beta power -1​

Answer 1

Answer:

alpha power -1 + beta power -1 = (alpha + beta)/alpha * beta

= (-5)/6

Answer 2

Answer

The value of α⁻¹ + β⁻¹ is -5/6

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

The quadratic polynomial is

• x² + 5x + 6
• α and β are the zeroes of the given polynomial

$\bf \large \underline{To \: Find : }$

• The value of α⁻¹ + β⁻¹ or 1/α + 1/β

$\bf\large \underline{Solution : }$

Given , α and β are the zeroes of the of the polynomial x² + 5x + 6

From the relationship of sum of zeroes and coefficients we have ,

$\sf\implies \alpha + \beta = \dfrac{-5}{1} \\\\ \sf\implies \alpha + \beta = -5\longrightarrow(1)$

And from the relationship of product of zeroes we have ,

$\sf\implies \alpha\beta= \dfrac{6}{1}\\\\ \sf\implies \alpha\beta = 6 \longrightarrow (2)$

$\sf \dag \: \: Dividing \: (1) \: by (2) \: we \: have$

$\sf\implies \dfrac{\alpha + \beta}{\alpha\beta}=\dfrac{-5}{6} \\\\ \sf\implies \dfrac{\alpha}{\alpha\beta}+\dfrac{\beta}{\alpha\beta} = -\dfrac{5}{6} \\\\ \sf\implies \dfrac{1}{\beta}+\dfrac{1}{\alpha}=-\dfrac{5}{6} \\\\ \sf\implies \dfrac{1}{\alpha}+\dfrac{1}{\beta}=-\dfrac{5}{6} \\\\ \implies\alpha^{-1} + \beta^{-1} = -\dfrac{5}{6}$