Question

How do you find the values of k that will make `kx^2 - 24x +16` a perfect square?

Answer 1

`k=9`

Explanation:

Notice that:

`(ax+b)^2 = a^2x^2+2ab+b^2`

Equating this with `kx^2-24x+16` we find:

`{ (a^2 = k), (2ab = -24), (b^2 = 16) :}`

So `k = a^2 = (-24/(2b))^2 = 12^2/b^2 = 144/16 = 9`

Answer 2

Consider the following factoring of a quadratic equation:

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

`0 = (x + 1)(x + 1)`

`x = -1`

This equation has only one solution.

The discriminant (`b^2 - 4ac`) tells us how many solutions/the type of solutions that a quadratic equation has. For your problem, we'll consider `y = 0`

As shown in the example above, in a perfect square trinomial there will only be one solution. This can be obtained by setting the discriminant to 0, and solving for a (k in our case), the term we don't know.

`(-24)^2 - (4 xx k xx 16) = 0`

`576 - 64k = 0`

`-64k = -576`

`k = 9`

Therefore, `k = 9`

Hopefully this helps!