The classroom was already busy when Anaya walked in.
Naina was sitting beside Ritu today.
The seat on Ritu's left was empty.
Anaya dropped her bag beside the desk and sat down.
The bell rang.
A few moments later, Aarav entered.
Several homework notebooks appeared on desks.
Aarav ignored them.
He picked up a piece of chalk.
"Any questions?"
A chair shifted.
Rishu stood up.
"Sir, in Quadratic Equations, if the discriminant already tells us the nature of roots, then why do we still find the roots?"
Aarav turned toward the board.
D = b² - 4ac
Then below it:
x = (-b ± √(b² - 4ac)) / 2a
"The discriminant tells us the nature of roots."
The chalk tapped the second formula.
"This gives the actual values of the roots."
Rishu sat down.
Another hand rose.
"Sir, if D becomes zero, are the roots always equal?"
Aarav wrote:
x = (-b ± √0) / 2a
Then below it:
x = -b / 2a
"If the discriminant is zero, both roots coincide."
The chalk stopped.
"So both roots are equal."
A student near the window stood.
"Sir, why is the empty set considered a subset of every set?"
Aarav wrote:
∅ ⊆ A
"Because there is no element in the empty set that can violate the definition of a subset."
The student nodded and sat down.
Another hand rose.
"Sir, if one element of the domain has two images, can it still be called a function?"
Aarav turned toward the board.
1 → 2
1 → 3
"One input."
The chalk pointed to the first number.
"Two outputs."
The chalk moved to the right.
"That is not a function."
The board slowly filled with symbols, conditions, and short examples.
Anaya's notebook filled with small notes.
"No more questions?"
Nobody stood.
Aarav nodded once.
"Classwork."
He turned toward the board.
The chalk touched the surface.
And the questions appeared.
Heads lowered.
Pages turned.
Pens moved.
The classroom settled into silence.
Anaya copied all five questions first.
Then she started.
2x^2-7x+3=0
D=(-7)^2-4(2)(3)
D=49-24
D=25
The next line stopped halfway.
A small mark crossed through it.
Another line appeared below.
Then another.
A few seats ahead, somebody turned a page.
Anaya moved to the next question.
3x^2-10x+3=0
D=100-36
D=64
x=\frac{10\pm8}{6}
The answer remained unfinished.
Her eyes moved back to the previous step.
Then to the question again.
A line disappeared beneath her pen.
Around the room notebooks continued filling.
Some students were already halfway down the page.
Others barely looked up from their work.
The scratching of pens never stopped.
2xsquare-8x+12=0
x^2-6x-2x+12=0
The pen paused.
She stared at the line.
Read it again.
Then drew a neat stroke through part of the solution.
Aarav began moving between the rows.
One desk.
Then another.
A notebook.
A rough copy.
A half-finished solution.
His hand paused near the corner of Anaya's desk.
The small keyring disappeared into his fingers.
ting.
Anaya's pen stopped.
Her eyes lifted.
The keyring swung lightly as Aarav moved toward the next row.
For a second her gaze remained on it.
Then shifted upward.
Toward Aarav.
A strange look crossed her face.
Gone almost immediately.
She lowered her head and continued writing.
4x^2-4x+1=0
D=16-16
D=0
She stared at the answer.
Looked at the question.
Then at the answer again.
The page remained open.
Nothing changed.
ting.
Her eyes moved again.
First the keyring.
Then Aarav.
That same odd expression appeared briefly.
Then she looked away.
Back to the notebook.
Several notebooks near the front were already turning pages.
One student had started checking answers.
Another had reached the last question.
Anaya finally moved down the page.
x^2-5x+k=0
One root is twice the other.
alpha=2\beta
The next line remained empty.
She wrote something beneath it.
Stopped.
Crossed it out.
Started again.
A few minutes later—
\alpha+\beta=5
She stared at the relation.
Then at the line above.
Then back again.
A mark cut through the page.
Another appeared below it.
ting.
Without thinking, her eyes lifted.
The keyring moved once.
Her gaze followed it.
Then settled on Aarav for a second.
That same strange expression crossed her face.
A moment later she returned to the notebook.
Half-finished calculations.
Crossed-out work.
Fresh attempts.
More crossed-out work.
Near the front, a notebook closed.
Then another.
A few students had already put their pens down.
Some were checking answers.
Others simply waited.
Anaya's eyes moved to the board.
Then back to her notebook.
Board.
Notebook.
Board.
Notebook.
The solution remained exactly where she had left it.
Aarav returned to the board.
Several heads lifted.
He picked up the chalk.
A solution began appearing line by line.
The classroom became completely silent.
Pens moved again.
Anaya's eyes followed the board.
Then her notebook.
Then the board.
Then her notebook.
Her pen moved quickly across the page.
A new step appeared beneath the old one.
Then another.
Then another.
Near the first bench, Aarav stopped.
"Done?"
Navya looked up.
Then quietly turned her notebook toward him.
He glanced at the page.
Nodded once.
Navya smiled and returned to writing.
The bell was only a few minutes away.
For the first time, Anaya looked around the room.
Several notebooks were already complete.
A few students had put their pens down.
Others were comparing steps.
Her eyes dropped back to her own notebook.
One answer had been corrected three times.
Another still carried crossed-out lines through the middle.
The last solution stretched across half the page.
Aarav picked up the chalk once more.
Immediately the room became alert.
One exercise appeared on the board.
Then another.
two complete exercises covered the board.
A brief silence spread across the room.
A few students stared.
One boy looked from the board to his notebook and back again.
Nobody said anything.
"Complete both exercises."
The bell rang.
Chairs moved.
Bags opened.
Students hurried to copy the remaining questions.
Voices slowly returned.
Anaya closed her notebook.
Slid it into her bag.
Then her eyes landed on the corner of the desk.
The keyring.
She picked it up.
The tiny bell moved lightly.
ting.
A second later it disappeared into her bag.
Around her, students were already leaving.
The board remained covered in Mathematics.