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

2 Answers
May 9, 2016

#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#

May 9, 2016

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!