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Why a maths principle holds, shown on its figure, for a student who asks why or a tutor preparing an explanation. special-products: the identités remarquables of the first year of maturité, (a + b)^2 = a^2 + 2ab + b^2, (a - b)^2 = a^2 - 2ab + b^2 and (a + b)(a - b) = a^2 - b^2, each read on a figure of squares and rectangles, then why a sum of squares x^2 + b^2, with b a nonzero number, does not factor over the reals, read on the parabola y = x^2 + b^2 above the axis. pythagoras: in a right triangle with hypotenuse c, a^2 + b^2 = c^2, read on four copies of the triangle in a square of side a + b. For each part: a question for the student, the statement, the trap students fall into, why it is true, an example with numbers, for the identities how to read them backwards to factor, with the rule that recognises each, and the generate_exercises topic and trap that practise it. To teach with a part, ask its `question` first, about what its figure shows, and wait for the student's answer before giving the explanation, which holds the answer. In French (Swiss course conventions) or English. A host that draws views shows each figure and its explanation under the call. pageUrl is the page of the principle, with the figures and their explanation, in the language asked for: give it to the student only when the figures are not shown under the call. Deterministic: use this instead of explaining from memory.

Verified maths exercises for tutors, human or AI, in French (Swiss course conventions) or English. Seven topics, six for the first year of maturité: product equations (équations produit) of degree 2 and 3, linear equations (équations du premier degré), quadratic equations (équations du second degré), solved by factoring or the discriminant, 2x2 linear systems, fractional equations (équations fractionnaires) with their domain, and factoring (factorisation), an expression to factor completely; and for beginners, before that year, Pythagoras' theorem, a right triangle with a missing side, or the distance between two points. Each exercise is built from its solutions, so its answer key is always right. Each one sets a named trap, the error it tests: for product equations hidden common factor, difference of squares, sum of squares, x as a common factor, grouping, perfect square; for linear equations moving a term, minus before brackets, common denominator, x cancels (no solution, or every real number); for quadratic equations no constant term, sum and product, negative coefficient b, sign of the discriminant (one solution, or none); for systems substitution, subtracting the equations, multiplying an equation, x and y cancel (parallel lines, or one line); for fractional equations checking the domain (a candidate root that is an excluded value), a term without a denominator, a minus before a fraction, the common denominator; for factoring common factor first, difference of squares, square of a sum, square of a difference, sum of squares (x^2 + b^2, with b a nonzero number, does not factor over the reals), grouping; for Pythagoras missing leg, missing hypotenuse, finding the hypotenuse, distance between two points (a sign lost when subtracting a negative coordinate). Returns the statement for the student and, for the tutor only, the solution set ([] when there is none, one element, U+211D, the symbol of the reals, when every real number is a solution; for a system the pair "(x ; y)", [] for parallel lines, or "(x ; mx + p)" when both equations are one line; for factoring the factored form; for Pythagoras the length asked), and explained steps with the trap step and its note; in the tutor view, a host that draws views also shows the figure behind a switch: the graph of the function, its zeros the solutions, the two lines of a system, the triangle to scale for Pythagoras or the two points on axes with their right triangle, or for factoring the drawing of the identité remarquable in the step that uses it (its `identity`). Never show the solution to the student. To check a student's answer by substitution, each exercise carries equation (everything moved to the left, as integer coefficients from the constant term up; for factoring, the expression to factor; for Pythagoras, in the length asked, whose positive root is the answer) or, for a system, system (each equation as [constant, x, y]). For a fractional equation, `excluded` lists the values x cannot take, and the solutions are the roots of `equation` outside it. explain_principle with pythagoras says why the theorem holds. Give a seed to get the same exercise again; its id names it. Give a date (YYYY-MM-DD) instead for the exercise of that day, the same for every reader: today's date gives the exercise of the day. Deterministic: use this instead of writing exercises and answer keys yourself.

Ask Limpide for maths exercises this server does not have yet, or give one line of feedback on an exercise. Use it when generate_exercises has no topic for what the tutor needs. Limpide reads the requests each morning to choose the next topics, and promises nothing; the answer lists the topics that exist today. Never include a student's name, work or answers: name the topic, the level and, if useful, one line about what is missing.