Question

For which value of k will the roots of `2x^2+kx+1=0` be real?

Answer 1

The answer is `k in ] -oo,-sqrt8 ] uu [sqrt8, oo[`

Explanation:

In order for the roots of a quadratic equation, to de real, the dircriminant `Delta>=0`

If the equation is `ax^2+bx+c=0`

the discriminant is `Delta=b^2-4ac`

`2x^2+kx+1=0`

`Delta= k^2-4*2*1>=0`

`k^2-8>=0`

`(k-sqrt8)(k+sqrt8)>=0`

We do a sign chart

`color(white)(aaaa)` `k` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-sqrt8` `color(white)(aaaa)` `sqrt8` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `k+sqrt8` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `k-sqrt8` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `Delta` `color(white)(aaaaaaaa)` `+` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+`

So, for `Delta>=0`

`k in ] -oo,-sqrt8 ] uu [sqrt8, oo[`