+972 Let $m$ be a real number. If the quadratic equation $x^2+mx+4 = 2x^2 + 17x + 8$ has two distinct real roots, then what are the possible values of $m$? Express your answer in interval notation.
+1950 First, let's combine all like terms and move all terms to one side.
We get
x2+(17−m)x+4>0
Let's note that if roots are real, then the descriminant must be greater or equal to 0.
We have
(17−m)2−4(1)(4)>0(17−m)2>16
From here, there are 2 ways to go. First, we have
17−m≥417−4≥mm≤13
and, we have
17−m≤−421≤mm≥21
This means our final answer is
m=[−∞,13]⋃[21,∞]
Thanks! :)
