Question

Si la suma de las inversas de las raíces de la ecuación: x^2-mx+1=0, es igual a la inversa de la suma de las raíces ¿qué valor asume «m»?

Answer 1

Let
`x_1, x_2` be the two roots. The claim is that

`(1)/(x_1)+(1)/(x_2)=(1)/(x_1+x_2)`

We can rewrite this expression as

`(x_1+x_2)/(x_1x_2)=(1)/(x_1+x_2)`

Now, recall that if the leading term is 1, then you can think of a quadratic equation as

`x^2-sx+p=0`

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

`x_1+x_2=m,\quad x_1x_2=1`

Substitute these values in the equation above to have

`(m)/(1)=(1)/(m)`

Which leads to

`m^2=1 \iff m=\pm 1`