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.
Step 6. Check the answer in the problem and make sure it makes sense.
Yes, 75+0.25\left(500\right)=200.
Step 7. Write a sentence that answers the question. Sergio and Lizeth can travel 500 miles and still stay on budget.
Taleisha’s phone plan costs her ?28.80 a month plus ?0.20 per text message. How many text messages can she use and keep her monthly phone bill no more than ?50?
Answer:
19
Step-by-step explanation:
KM=4
KN=9
9-4=5
MN=5
NL=14
5+14=19
