Question 1209496
.
(a) Express x^2 + 6kx + 144 in the form of (x+p)^2 + q. 
(b) Find the range of values of k suck that the x^2 + 6kx + 144 is positive for all values of x.

[the answer for (a) is solved already, I need the second part please.]
[the answer for (b) is -4<k<4 if you need it]
Thank you!
~~~~~~~~~~~~~~~~~~~~~~~~



I will work for part (b), ONLY.



<pre>
Consider this polynomial  x^2 + 6kx + 144  and find the discriminant for it

    d = b^2 - 4ac = (6k)^2 - 4*1*144 =  = 36k^2 - 4*144.


The polynomial  x^2 + 6kx + 144  is positive for all real values of x
if and only if the discriminant is negative D < 0

    36k^2 - 4*144 < 0,  or  36k^2 < 4*144,  or  k^2 < {{{sqrt((4*144)/36)}}} = {{{sqrt(4*4)}}} = 4.


Taking the square root of both sides, we get

    |k| < 4,  or, which is the same,  -4 < k < 4.


<U>ANSWER</U>.  The range for k is  -4 < k < 4.
</pre>

Solved.