My subject turned into a beautiful girl

The wind lashed the rain against the glass in bursts, cascading down like a waterfall. The osmanthus tree outside the window swayed violently, trying to avoid the storm, but still couldn't escape breaking branches and falling leaves. Lightning flashed intermittently, shattering the inky blackness of the sky, making everything seem as if the end of the world was imminent.

An Yi has always loved this kind of weather. Whenever he encounters it, he always feels as if the pent-up frustration in his heart has been blown away by the strong wind, making him feel exceptionally refreshed and almost wanting to shout something out loud.

However, he was never one to seek the limelight, so he would often choose to silently gaze out the window, lost in thought, imagining himself standing up to save the world if the end of the world really came and everyone was in a panic.

In these scenes, his superpowers change with age, from a light stick to the Emperor Armor and then to Iron Man's suit, with no two being the same.

The only thing that remains the same is his cool and collected demeanor as he leaves without saying a word after each world-saving mission... So awesome!

This time, however, he paid no attention to the scenery outside the window or imagined any scenes. Instead, he devoted himself to learning how to operate the CASIO graphing calculator from his older brother.

"Just enter all the functions you want to draw, then press F6."

Xia Qing inputs the functions Y₁=-x²; Y₂=-x²-1, and after pressing the F6 key, the image is generated, and the two quadratic functions appear on the screen, one blue and one red.

"Try adjusting these two functions to make them intersect, and draw an eyebrow...wait, use your pen and scrap paper."

"Brother, pen and ink."

An Yi respectfully presented the ballpoint pen from his pencil case, a pen that was only used for exams. This was a great opportunity to have the top student "bless" it, and he couldn't miss it.

"Please wait a moment for the paper!"

As he spoke, he pulled a blue plastic-covered folder from the wall of books, opened it, and then took out a blank math answer sheet from the transparent plastic inner pages.

"oops?"

Xia Qing's eyes lit up. "Ninety-nine percent! A rare find!"

"Of course." An Yi nodded solemnly.

For most high school students, the phrase "anything can be used as a draft" is no exaggeration. It includes everything from the gaps in test papers and textbook pages to the back of daily quizzes, palm-sized sticky notes, and even tissues, toilet paper rolls, and the palm or back of one's hand in emergencies…

It is precisely under this premise that those completely blank draft papers are all absolutely strategic resources!

Strategic resources also vary in quality, primarily determined by paper quality and writing feel. For example, the special scratch paper provided for exams is too thin; it's barely adequate for NPCs (National Standard Examination Boards). A4 printing paper is quite good and can be given to top-tier institutions. Blank answer sheets, however, due to their thickness, size, writing feel, and scarcity, undoubtedly hold a perpetually top-tier status.

Among them, the math answer sheet is known as the "god among gods" because it has the most blank space and the highest space utilization rate!

The significance of strategic resources lies in storing them away unused in daily life until "wartime." For example, a difficult problem that requires extensive drafts; a long evening study session involving countless rounds of Gomoku; or... right now.

An Yi had a few clever ideas. On one hand, since his elder brother had already taught him the "Burning Technique," how could he not offer him the best treatment? On the other hand, the answer sheet felt really good to the touch. What if his elder brother got into the swing of things while writing and taught him a little more?

Xia Qing took the math answer sheet, first running her hand over it to feel the texture, and then began to write. Two smooth lines of ink met on the paper, outlining a small, curved eyebrow. (See picture ↓)

"Drawing it like this is fine."

She paused, whether unable to resist the temptation of such offerings or having planned it all along, she said, "I'll draw you a few more patterns, and you can copy them? That way you'll learn more efficiently."

"Okay, okay..."

An Yi agreed, but her attention quickly shifted to figuring out how to "embellish" the two quadratic functions.

Instead of trying it directly on the graphing calculator Xia Qing handed him, he first took his usual pen and found a gap on his nearly used draft paper to roughly draw a before-and-after comparison diagram.

Such a transformation is not difficult. Even if An Yi's foundation in the properties of function graphs is not solid, he can easily come up with a clear idea.

But he did not develop any contemptuous feelings because of this.

Over the years, he has personally experienced and come to understand a principle many times—

The beginning of the story is always very gentle.

When he first started learning math, physics, and biology, he thought it was quite easy, so he naturally relaxed his vigilance, became distracted, and you can tell what happened by looking at his current grades!

As for chemistry? For An Yi, chemistry was never a gentle subject from the start.

"First, change the value of 'a' to alter the size of the opening, creating different openings so that the two quadratic functions intersect within a small range. Then adjust the axis of symmetry, following the function translation rules: add to the left, subtract to the right, add to the top, and subtract to the bottom. That's the general process. Finally, adjust the detailed values ​​based on the image..."

After figuring out what transformation to make, An Yi tried it out on the graphing calculator. During this process, his greatly enhanced calculation ability was basically useless, because when translating the function, he directly expressed it as a quadratic function vertex form, i.e., y=a(xh)²+k, which was more intuitive and convenient to adjust.

After several adjustments, he obtained a new function expression:

Y₁=-0.15x²; Y₂=-0.08(x+0.4)²-0.6

It looks almost identical to the pattern drawn by my older brother. (See picture below ↓)

The next step is to calculate the intersection point AB of the two quadratic functions, which is the coordinate of the beginning and end of the eyebrow. Then, assign values ​​to the domain to extract the required range, and you can declare success.

But An Yi still felt unsatisfied looking at the images.

Xia Qing said this is a "relatively simple version," so what are the differences in the more complicated version?

An Yi was never one to create trouble for himself, especially when it came to his studies; taking things easy was his consistent principle.

But the situation is slightly different now.

When people have nothing to do, they won't walk into weedy areas because getting cut by the grass will cause itching and getting bitten by insects will cause pain. But if they have a straight and sturdy stick in their hand, they will subconsciously go there and start swinging the stick.

The boy wields a stiff, straight stick, bending all the grasses for miles around!

First, he identified the area that needed modification. The combined graph of the two functions already looked like a real, small, curved eyebrow; the problem lay in that small section.

The eyebrow on the right side of the image should be rounded, not sharper than the eyebrow tail on the left. Furthermore, the two should be roughly level, not one higher than the other.

"An arc, so let's use the standard equation of a circle to represent it? The condition that this circle needs to satisfy is that it is tangent to two quadratic functions, and the other endpoint B of the intersection with the two quadratic functions is about the same height... Then, based on the intersection point, we can take the domain and obtain the arc."

As An Yi pondered this, the pen in her hand unconsciously spun faster and faster until it suddenly stopped.

He already understood the detailed steps and thought process; the next step was to perform the calculations.

The standard equation of a circle is (xa)² + (yb)² = r². He traced it on a piece of draft paper and decided to set the value of a to 2. This gave him the equation (x-2)² + (yb)² = r². He then used this equation to solve the problem by combining it with a quadratic function.

Setting up a system of equations is simple. The equation of a circle is tangent to a quadratic function if two conditions are met simultaneously: they have a common point and the derivatives at the common point are equal, which means the slopes of the tangent lines are the same.

Based on these two mathematical conditions, let the point of tangency between the circle and Y1 = -0.15x² be (x₁, y₁), where the slope of the circle at that point is derived implicitly using the derivative of the function:

2(x−2)+2(y−b)y′=0→y′=-（x-2）/（y-b）

Equal slopes yield -0.3x₁ = -(x₁ - 2) / (y₁ - b)

We obtain the following system of equations: y₁=-0.15x₁²; (x₁−2)²+(y₁−b)²=r²; -0.3x₁=-（x₁-2）/（y₁-b）

Similarly, the system of equations relating the circle and Y₂ = -0.08(x₂ + 0.4)² - 0.6 is as follows:

y₂=-0.08（x₂+0.4）²-0.6；(x₂−2)²+(y₂−b)²=r²；−0.16(x₂+0.4)=-（x₂-2）/（y₂-b）

Combining these six equations forms a system of simultaneous equations, with a total of 6 unknowns and 6 equations.

In the past, An Yi would have stopped there; he could list out the system of equations, but he didn't have the ability to solve them.

But now...

The weeds are right in front of us, and the stick is in our hands. If we don't sweep them now, when will we?

[Computational ability] is not just limited to simple numerical operations; it also includes algebraic operations!

An Yi put pen to paper, and numbers and symbols appeared on the paper without any thought, so fast that it was like copying rather than calculating.

His proposed steps were very simple.

第一步，将y₁=-0.15x₁²与y₂=-0.08（x₂+0.4）²-0.6分别代入-0.3x₁=-（x₁-2）/（y₁-b）和−0.16(x₂+0.4)=-（x₂-2）/（y₂-b），将x₁与x₂基于b的表达式解出。

The second step is to substitute y₁y₂ into (x₁−2)²+(y₁−b)²=r²=(x₂−2)²+(y₂−b)², and then substitute the expressions for x₁ and x₂ based on b into the equation, thus obtaining an equation containing only b.

The third step is to solve the equation to obtain the value of b, after which the solution for r will be readily available...

Just as An Yi finalized the steps, a semi-transparent light curtain appeared—

You want to play a game of untying knots with math. This is a game you've been playing together from time to time since you met four years ago, and it's continued to this day.

You've evolved this game into many different forms, and there are many knots that can be perfectly untied in various different ways... Mathematics has always been very interested in this.

Is untying a knot the same as solving an equation?

This correspondence is not hard to guess, after all, the time point has been given, and fourth graders are just learning about equations!

But what the hell is that below?!

An Yi lowered her gaze, her heart skipped a beat.

You think untying this knot is easy. Although Math senses something's wrong, given how angry you were with her before, a little revenge seems reasonable. She's not going to say it aloud, but instead ambiguously lets you try, wanting to see you fail.

What does it mean when "mathematics senses something is wrong"?

An Yi had a very bad feeling. At this moment, he clearly realized that the "straw" he had brought to test the stick might be a steel bar disguised as a stick.

But at the same time, he also knew very well that now, let alone steel bars, even if it were replaced with uranium rods from a nuclear reactor, he would have to bite the bullet and sweep them up with a stick!

After all, mathematics is already prepared to "watch the joke".

The nature of mocking someone depends on the context. If it happens in public, in full view of others, it's considered malicious ridicule and can be quite hurtful. However, if it's limited to between two people, or is a small act of "revenge" during the initial reconciliation after an argument…

These are small, intimate interactions that can enhance relationships!

If the setting weren't so inappropriate, An Yi would almost want to wriggle with happiness at the thought. Whether she wriggled like a puppy wagging its tail or squirmed like a maggot on the bed didn't matter... After all, she was so beautiful, right?

In short, An Yi's stick still fell on the pile of what looked like "straw" but was actually steel bars or uranium rods.

At the end of the first step, he realized something was wrong.

Because he was facing a cubic equation in two variables.

In this case, how do we solve for the b-based expressions for x₁ and x₂?

But he didn't stop there. Instead, he chose to leave the algebraic calculations to his "instinct." That is, just like when he first tested his calculation ability, he input the equation, the desired answer, and then got the result.

The beauty of numbers was on full display in the next moment.

An incredibly complex algebraic expression for cube roots appeared in his mind!

An Yi was stunned for a moment, and then she was both delighted and surprised when she realized what was happening.

He was happy because he discovered that his stats after the upgrade were really high, allowing him to fly bricks with great strength.

As for the surprise... he already roughly knew just how hard the "straw" was.

Instead of writing down the algebraic expression for the cube roots of x₁ and x₂ based on b, he chose to proceed to the next step on the draft paper first.

Sure enough, after barely completing the second step, An Yi was completely stuck on the third.

This time, even his decision to rely on "instinct" for algebraic calculations was futile. No final answer emerged, and he even felt a slight burning sensation in his head…

"Is this my current numerical limit?"

An Yi paused, pen in hand, and decisively chose to give up.

But at the same time, a question arose in his mind—

Why can't I solve it?

Sometimes things just happen by chance. Just as he was wondering, a light, casual remark came from the side.

That's enough.

Xia Qing, who had been silently observing An Yi while drawing, shook her head slightly, as she noticed An Yi starting to increase the difficulty of his drawings.

"This equation has no radical solution."

"what?"

An Yi was taken aback. "No solution? That's impossible!"

"It's not that there's no solution, but rather that there is no 'radical solution'."

Xia Qing wrote the words "radical solution" on the draft paper.

"You can understand it as any solution that can be written out using a finite number of additions, subtractions, multiplications, divisions, and expressions with radicals is a radical solution."

The radical solution is exact, but this equation does not have a radical solution; only a numerical method can yield an approximate solution. Therefore, you should stop using algebraic calculations here, because continuing with this method, even a supercomputer, will not be able to solve it.