Let [tex]x_1, x_2[/tex] be the two roots. The claim is that
[tex]\dfrac{1}{x_1}+\dfrac{1}{x_2}=\dfrac{1}{x_1+x_2}[/tex]
We can rewrite this expression as
[tex]\dfrac{x_1+x_2}{x_1x_2}=\dfrac{1}{x_1+x_2}[/tex]
Now, recall that if the leading term is 1, then you can think of a quadratic equation as
[tex]x^2-sx+p=0[/tex]
i.e. the linear coefficient is the opposite of the sum of the roots, and the constant term is the product of the roots. In other words, we have
[tex]x_1+x_2=m,\quad x_1x_2=1[/tex]
Substitute these values in the equation above to have
[tex]\dfrac{m}{1}=\dfrac{1}{m}[/tex]
Which leads to
[tex]m^2=1 \iff m=\pm 1[/tex]
