Question

Se a equação x2 + 2x – 8 = 0 tem as raízes a e b, então o valor de (1/a – 1/b)2 é:

Answer 1

$\mathsf{{x}^{2}+2x-8=0}\\\mathsf{\Delta={2}^{2}-4.1.(-8)=4+32=36}$

$\mathsf{x=\dfrac{-2\pm6}{2}}\\\mathsf{x_{1}=\dfrac{-2+6}{2}=\dfrac{4}{2}=2}\\\mathsf{x_{2}=\dfrac{-2-6}{2}=-\dfrac{8}{2}=-4}$

$\mathsf{a=2~~b=-4}$

$\mathsf{{ \left(\dfrac{1}{2}-\dfrac{1}{-4} \right)}^{2}={ \left(\dfrac{1}{2}+\dfrac{1}{4} \right)}^{2}} \\\mathsf{ { \left( \dfrac{2 + 1}{4} \right)}^{2} = {\left(\dfrac{3}{4} \right)}^{2}} = \huge \boxed{\boxed{\mathsf{ \dfrac{9}{16} }}}$