Question

If the equation `x^2-kx+1=0` has no real roots the what is true for k from the following and how ? A k>-2 B. k<2 C. -2<k<2 D. None

Answer 1

The answer is option `=(C) -2 < k < 2`

Explanation:

For the quadratic equation `x^2-kx+1=0`

To have no real roots, the discriminant `Delta<0`

`a=1`

`b=-k`

`c=1`

Therefore, the discriminant is

`Delta=b^2-4*a*c=(-k)^2-4*1*1=k^2-4`

So,

`k^2-4<0`

`(k+2)(k-2)<0`

Make a sign chart to solve this inequality

`color(white)(aaaa)` `k` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `+2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `k+2` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `k-2` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+`

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

Finally,

`Delta<0`, when `k in (-2,+2)`