SOLUTION: Express x^2 + 6kx + 144 in the form of (x+p)^2 + q. 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

Algebra ->  Equations -> SOLUTION: Express x^2 + 6kx + 144 in the form of (x+p)^2 + q. 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       Log On


   

Question 1209496: Express x^2 + 6kx + 144 in the form of (x+p)^2 + q.
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 Thank you!
Answer by ikleyn(52812) About Me  (Show Source):
You can put this solution on YOUR website!
.
(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 Thank you!
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I will work for part (b), ONLY. 


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%28%284%2A144%29%2F36%29 = sqrt%284%2A4%29 = 4.


Taking the square root of both sides, we get

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


ANSWER.  The range for k is  -4 < k < 4.

Solved.