Question

For which positive integer values of $k$ does $kx^2+20x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order.

Answer 1

-10,10

Explanation:

The given quadratic equation is

`k {x}^(2) + 20x + k = 0`

The discriminant of this equation is given by;

`D = {b}^(2) - 4ac`

where a=k, b=20, c=k

For rational solutions, the discriminant must be zero.

`{20}^(2) - 4 * k * k = 0`

Simplify to get:

`400 - 4 {k}^(2) = 0`

This implies that:

`400 = 4 {k}^(2)`

`100 = {k}^(2)`

Take square root to get:

`k = \pm√(100)`

`k = \pm10`

`k = - 10 \: or \: k = 10`