Algebra 2

Quadratics

ladderAIAlgebra 2written with an agent#c8f181996d

Adam Zettel@adamSep 22, 2026

The quadratic formula, derived

A garden has a perimeter of 10 m and an area of 6 m². How long are its sides? The answer needs $x^{2} - 5 x + 6 = 0$ , and solving it by completing the square gives the quadratic formula.

1Complete the square+1 function · +1 test
2Solve from the vertex+1 function · +1 test
3When there are no real roots+1 function · +1 test

✓ 3 tests passon 3 rungs, each rerunning every earlier test · checked when the site was built#algebra #quadratics #ladder

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problemAIAlgebra 2written with an agent#caa1277eba

Adam Zettel@adamSep 22, 2026

Solve the quadratic

A ball is thrown straight up at 4 m/s from 1 m above the ground. When does it land? Write roots(a, b, c), the two solutions of $a x^{2} + b x + c = 0$ , smaller first.

x^2 - 5x + 6 = 0 gives 2 and 3
-5t^2 + 4t + 1 = 0 gives -0.2 and 1, smaller first

✗ 2 tests to solvethe unfinished functions run, and the failing tests show what they must compute1 solution #algebra #problem

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solutionAI solves Solve the quadratic Algebra 2written with an agent#eda8efc005

Adam Zettel@adamSep 22, 2026

The quadratic formula, applied

The ball thrown up at 4 m/s from 1 m lands after 1 s. The quadratic formula gives that, but for $a < 0$ its minus sign gives the larger root, so the solution writes the roots as $h \pm w$ with $w \geq 0$ .

✓ 2 tests passthe problem's tests, unchanged, among them · checked when the site was built#algebra #solution

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Systems of equations

problemAIAlgebra 2written with an agent#ea7a9e6867

Adam Zettel@adamSep 22, 2026

Two lines cross

(a d b e) (x y) = (c f)

Two straight lines, $a x + b y = c$ and $d x + ey = f$ , meet at one point unless they are parallel. Write solve2 for that point, and determinant for the single number that tells you whether it exists.

x + y = 5 and x - y = 1 cross at (3, 2)
2x + 3y = 12 and x - y = 1 cross at (3, 2) too
3x - 2y = 4 and 5x + 4y = 14 cross at (2, 1), and the point satisfies both
the determinant is zero for parallel lines and -2 for x + y, x - y

✗ 4 tests to solvethe unfinished functions run, and the failing tests show what they must computeno solutions yet #algebra #linear-systems #problem

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Exponentials and logarithms

demoAIAlgebra 2written with an agent#56bea7ad50

Adam Zettel@adamSep 22, 2026

Doubling time

t_{2} = \frac{ln 2}{ln ( 1 + r )} \approx \frac{70}{100 r}

Something growing 7% a year doubles in $ln 2/ ln 1.07 = 10.24$ years, not $100/7 = 14.3$ : growth compounds. The logarithm is the tool that turns "how long until" into a division, and the rule of 70 is that division with $ln 2 \approx 0.7$ and $ln (1 + r) \approx r$ .

Growth at 7% a year, compounding and not

1 + 0.07 t

✓ 4 tests passchecked when the site was built2 forks #algebra #exponentials #logarithms

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problemAIAlgebra 2written with an agent#275bf07cad

Adam Zettel@adamSep 22, 2026

Compound interest

P r n times a year, t years A = ?

$1000 at 5% a year becomes $1628.89 in ten years if the interest is added once a year, and $1647.01 if it is added monthly. Write balance to get both from the same four numbers, and years_to_double to say how long doubling takes.

1000 at 5% for 10 years, compounded once a year, is1628.89
compounded monthly it is $1647.01
at 5% a year the money doubles in 14.21 years, and the balance then is 2P
no time, no interest: the balance is the principal

✗ 3 tests to solve1 pass already · the unfinished functions run, and the failing tests show what they must computeno solutions yet #algebra #exponentials #problem

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Sequences and series

demoAIAlgebra 2written with an agent#337924f62b

Adam Zettel@adamSep 22, 2026

The geometric series adds up

S_{n} = a \frac{1 - r ^{n}}{1 - r} \to \frac{a}{1 - r}

$1 + \frac{1}{2} + \frac{1}{4} + \dots$ never reaches 2, and the partial sums say exactly how far short: $S_{n} = 2 - 2^{1 - n}$ . The general fact is $S_{n} = a \frac{1 - r ^{n}}{1 - r}$ , and when $∣ r ∣ < 1$ the $r^{n}$ goes to zero and the whole series is $a / (1 - r)$ . The program adds the terms one by one and checks the formula at every $n$ .

Running sums of a geometric series

r = 1/2, limit 2r = 0.9, limit 10

✓ 5 tests passchecked when the site was built#algebra #sequences #series

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Quadratics, systems of equations, exponentials and logarithms, and sequences and series, taught as programs that run and test their own answers. Read a ladder or a demo, then solve the problem after it.

Every post here is one program. A demo shows a fact and tests it. A ladder derives a formula in steps, called rungs, each with its own tests. A problem is a program with unfinished functions and tests that say what they should return; you solve it by filling in the functions until the tests pass, either inside the post or in the post editor. Solutions can be posted too, so worked answers collect here as people solve the problems.

7 posts · 3 problems

How to read it

Read the posts from top to bottom. On a problem, press Solve it to open its program in the card, fill in the functions, and press Run.