You did it right (almost, I got 21 instead of 19) but didn't finish. You need to show your discriminant is never negative.
x² + (p+1)x = 5-2p
x² + (p+1)x +(2p-5) =0
Real roots mean a positive (or at least non-negative) discriminant:
D = b² - 4ac = (p+1)² - 4(1)(2p - 5) = p² + 2p + 1 - 8p + 20
D = p² - 6p + 21
It's not totally obvious that D>0; we prove that by completing the square by noting
(p-3)² = p² - 6p + 9
so
p² - 6p = (p-3)² - 9.
D = p² - 6p + 21
D = (p-3)² - 9 + 21
D = (p-3)² + 12
Now we clearly see D>0 always because the squared term can't be negative, so D is always at least 12. We always get two distinct real roots.
Answer:
15 (I think)
Step-by-step explanation
6 x 25 = 150
20% of 150 = 30
10% of 150 = 15
30 - 15 = 15
<em>Sorry if this is wrong.</em>
<em>Bye, have a great day/night.</em>
To find f(g(x)), we'll treat g(x) as if it were the x in f(x).
g(x) = x + 3
f(g(x) = (x+3)^2 - 1
Subtract 6 from 10 and your answer is x<4
