Real and Imaginary Roots

May 15, Friday, classroom of Class 2, Grade 11, Nangang Second High School.

After the last class of the morning, Chen Xiaoting quietly said to Shen Qi, "Shen Qi, want to go see a movie this Sunday?"

Shen Qi shook his head. "A movie this Sunday? I'm busy! You sure picked your moment. Did you do that on purpose?"

Chen Xiaoting glanced around like a thief and chided him, "Hey, could you keep your voice down?"

Shen Qi lowered his voice a little. "I really am busy this Sunday. I'm taking the Provincial High School Mathematics League, and I've already signed up."

"Oh, I see. Then the math competition definitely comes first. We'll go another time, some other day." Chen Xiaoting relaxed, though she also felt a little disappointed. "But Sunday's the only time I have a few free hours. Who knows when I'll be free again. Shen Qi, I'm really grateful that you tutor me in math every day. I've gotten so much better at solving math problems. I genuinely wanted to treat you to a movie, or a meal or something, to show my appreciation."

"We're classmates, and you're my deskmate. Don't say such tacky things. It hurts our friendship."

"All right, I was being tacky. Good luck, and come back victorious from the math competition!"

"Chen Xiaoting, what did you just say? You'd treat me to a meal? Then take me out for a simple lunch."

"..."

Shen Qi and Chen Xiaoting went to a chain Chinese restaurant outside the school. It was lunchtime, and the place was packed, mostly with students from nearby.

Nangang Second High School certainly had a cafeteria, but the food there was awful. Students either went home for lunch or ate somewhere outside the school.

Shen Qi and Chen Xiaoting joined the queue to order. Chen Xiaoting took the lead and stood in front of Shen Qi. She was treating him to lunch.

"Shen Qi, there are three set meals: spicy chicken leg rice with Overlord Flower Dragon Bone Soup, beef brisket and potato rice with American Ginseng Black Chicken Soup, and salted fish and eggplant rice with American Ginseng Black Chicken Soup. Which one do you want? I can order on my phone. If we pay together, we'll save five yuan." Chen Xiaoting tapped at her phone and asked him.

"Don't rush to order. There's a bug here." Shen Qi had been watching the restaurant's order information screen.

Chen Xiaoting looked confused. "A bug?"

The line moved forward, and Chen Xiaoting reached the counter.

"Welcome, miss! What can I get you?" the young male employee asked cheerfully.

Chen Xiaoting turned to Shen Qi. "I'll have the beef brisket and potato rice with American Ginseng Black Chicken Soup set meal. What about you?"

Shen Qi stepped up to the counter and said, "Your spicy chicken leg rice with Overlord Flower Dragon Bone Soup set meal costs 30 yuan, your beef brisket and potato rice with American Ginseng Black Chicken Soup set meal costs 28 yuan, and your salted fish and eggplant rice with American Ginseng Black Chicken Soup costs 25 yuan. These are really combinations of three main dishes and two soups."

The young employee smiled. "That's right. You're great at math, aren't you? You two are together, right? Which set meal would you like? There's a discount if you pay by phone."

Shen Qi continued unhurriedly, "According to your menu, the spicy chicken leg rice costs 22 yuan on its own, while the beef brisket and potato rice and the salted fish and eggplant rice cost 20 yuan and 15 yuan respectively."

The young employee still wore a professional smile, but he was getting impatient. "Would you like to order just a main dish without soup?"

"No. My classmate and I came here mainly for the soup. Your soups are pretty good."

"Then which set meal would you like?"

"Before I order, I need to figure something out: the prices of your two soups. You haven't listed them, but I can solve for the prices of the Overlord Flower Dragon Bone Soup and the American Ginseng Black Chicken Soup with a simple system of equations." Shen Qi looked at the menu screen again. He had already worked out the answer in his head.

"..." The young employee stopped smiling and thought, What the hell is wrong with you?

Chen Xiaoting was at a loss. Shen Qi, I'm just treating you to lunch. Don't make a scene!

Shen Qi explained seriously, "Let's say the Overlord Flower Dragon Bone Soup costs x yuan, and the American Ginseng Black Chicken Soup costs y yuan. That gives us three linear equations, which form a system of linear equations."

"After some fairly simple calculations, I got a value for x: the Overlord Flower Dragon Bone Soup costs 8 yuan. Okay, there's no problem with that."

"But what's interesting is that y has two possible values: 8 and 10."

"So your American Ginseng Black Chicken Soup can cost either 8 yuan or 10 yuan per bowl. The same y can't have two real roots in a system of linear equations. That raises another question: does your American Ginseng Black Chicken Soup have an imaginary root? What's the truth? Please tell me. As a customer, I have the right to know."

"I... I..." The young employee was at a loss for words. He cursed inwardly, Tell you my ass! If I could solve for an imaginary root, would I be stuck here smiling and selling fast food?

"Holy crap, a top student!"

"We've caught a wild top student!"

"Hey, what school are you from, Second High or Seventh High? Let's be friends and get together! Should I scan your code, or should you scan mine?"

There were lots of students waiting in line, and Shen Qi was instantly surrounded by a crowd.

"Pfft." Chen Xiaoting, who had been confused moments ago, snickered. Shen Qi did seem to have a point. Why did the same soup lead to two different prices? What was the truth? Shen Qi was remarkably observant. Most people wouldn't pay attention to such a detail, but he had noticed it.

"Yeah, why does your soup have two real roots?"

"That's not scientific!"

"Are you a shady shop?"

"We demand a reasonable explanation!"

"Hello, is this the Consumer Protection Association? There's a shady shop at 233 Tianhe East Street. I want to report it!"

"To report something, call 110? Then please transfer me to 110."

The students gathered around started clamoring, and the area in front of the counter descended into chaos.

"Students, please calm down!"

"Relax! Be rational!"

"Come on, everyone. You're all educated people!"

The employees scrambled to keep things under control. None of them could answer Shen Qi's math question, so all they could do was try to placate the rowdy kids.

At this critical moment, the manager stepped forward. "Our shop doesn't have imaginary roots, and each soup has only one real root! The display is wrong. The prices for the set meals are incorrect!" A person who could become a manager was, after all, better educated.

"Manager, actually, all I want is an explanation in terms of an Abelian group. Do you understand?" Shen Qi liked talking to professionals.

Go to hell with your Abelian group. I don't understand, and are high school students always this irritating these days? The manager couldn't answer Shen Qi's question, but he'd figured out what had caused the commotion: the American Ginseng Black Chicken Soup was off by two yuan. In other words, they'd been caught out. It was frustrating, to say the least. Even the Industry and Commerce Bureau didn't care about this—why the hell was a high school student making a fuss?

"All the soups in our shop have only one real root, and they all cost 8 yuan! The listed price for the salted fish and eggplant rice with American Ginseng Black Chicken Soup is wrong. It isn't 25 yuan; it's 23 yuan! Anyone who bought this set meal can show us the receipt and get a refund of 2 yuan!" The manager reacted quickly and immediately put an emergency measure in place.

"That explanation is just about reasonable." Shen Qi nodded, then said to Chen Xiaoting, "Let's both order the salted fish and eggplant rice with American ginseng black chicken soup. With the price adjustment and the discount, we'll save 9 yuan in total—that brings it down to 82% of the original price. I'd feel awkward letting a girl treat me to a meal, so I'll save you as much as I can."

"All right. You really went all out, bringing up real and imaginary roots just to save a few yuan. Shen Qi, you're something else!" Chen Xiaoting scanned the code on her phone and bought two salted fish and eggplant rice set meals.