Collection of exercises on systems of equations using substitution, elimination and reduction methods. Includes systems with 2 and 3 variables, special cases (indeterminate and inconsistent systems) and some simple nonlinear systems.
Solved exercises on second-degree equations: from the basics to the most complex cases. Learn how to use the discriminant (\( \Delta \)), the complete formula and the reduced formula with step-by-step solutions.
Guide to Solving Rational Equations. Step-by-step guided exercises: from determining the domain to clearing the denominators. A comprehensive collection to understand when a solution is valid or must be rejected as extraneous.
Solved Exercises on Quadratic Inequalities. A guided walkthrough of the sign analysis method. Learn how to move from the associated equation to the final solution set in a few straightforward algebraic steps.
Exercise 1 — 25/03/2026 — level ★★☆☆☆
\[ 3x^2 + 6x \]
\[ 3x(x + 2) \]
A progressive collection — from basic numerical examples to parametric identities — designed to master the fundamental laws of exponents.
Given two polynomials \(A(x)\) and \(B(x)\ne 0\), there exist unique polynomials — the quotient \(Q(x)\) and the remainder \(R(x)\) — such that:
\[A(x)=B(x)\cdot Q(x)+R(x),\qquad \deg R(x)<\deg B(x).\]
Exercise 1 — level ★☆☆☆☆
\[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=2x+1 \]
The function is bijective.
Radicals: definition, existence conditions, fundamental properties, simplification, operations and rationalisation. Examples and step-by-step solved exercises.
First-Degree Inequalities: Solved Exercises and Basic Rules. A practical guide to solving linear inequalities step by step.
Sets are all around us: the set of students in your class, the set of songs on your favourite playlist, the set of even numbers. But what happens when these sets meet? How can we combine them, compare them, or pull them apart?
Numbers are the universal language of mathematics. From simple counting of objects to the most complex equations of modern physics, numbers accompany us in every aspect of daily life and scientific research.
A second-degree inequality is an algebraic expression that establishes an order relation between two terms containing a second-degree variable. It can be written in the form:
This table collects the fundamental rules of differentiation.
The circle is the locus of points in the plane that are at a constant distance from a fixed point, called the center. This constant distance is called the radius.
The parabola is the locus of points in the plane for which the distance to a fixed point (focus) equals the distance to a fixed line (directrix).
The study of the relative positions of two lines is one of the fundamental topics in plane analytic geometry.
The projection of a point onto a line represents one of the fundamental concepts in analytic geometry.
A first-degree inequality is an algebraic expression that establishes an order relation between two terms containing a linear variable. It can be written in the form:
\[ a x + b \leq 0 \quad \text{or} \quad a x + b \geq 0 \]
Bernoulli's inequality, stated by the Swiss mathematician Jacob Bernoulli in 1689, is of fundamental importance because it allows us to establish upper and lower bounds for exponential and polynomial functions.
Numerical sequences and limits of sequences are fundamental concepts in mathematical analysis.
The Stolz-Cesàro theorem provides a useful tool for computing the limit of a ratio of sequences. It is particularly useful when the denominator tends to infinity and the computation of the limit is not immediate.
Cauchy's theorem is a fundamental result that extends Lagrange's theorem by introducing a relationship between two functions.
On this page we will see how to calculate the derivative of the logarithm with base \( b > 0 \) using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \):
Monotonic sequences (both increasing and decreasing) enjoy a very important property: they always have a limit, finite or infinite.
In mathematical analysis, a sequence is a rule that associates to each natural number \( n \in \mathbb{N} \) an element \( a_n \) belonging to a set \( X \).
The Lagrange Theorem, also known as the Mean Value Theorem, is a fundamental result in mathematical analysis.
Rolle's Theorem is a fundamental result in the study of continuous and differentiable functions.
The Weierstrass Theorem states that a continuous function defined on a closed and bounded interval necessarily attains a maximum value and a minimum value.
To thoroughly understand the properties of logarithms, we will start from their definition. From here, we will demonstrate step by step the main rules that allow us to simplify and manipulate logarithmic expressions.
Let us see how to calculate the derivative of the sine and cosine functions, using the limit of the difference quotient and fundamental trigonometric identities.
The mode is one of the simplest and most useful measures of central tendency for understanding the distribution of a dataset. It represents the value that appears with the highest frequency within a dataset.
The arithmetic mean, also called simply the mean, is one of the most widely used measures of central tendency in statistics.
On this page we will see how to calculate the derivative of the natural logarithm using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:
Limit operations are of fundamental importance because they allow us to calculate the limit of a sum, of a product, or of a quotient by deducing it directly from the limits of the individual sequences
We begin with the derivative of the tangent \( f(x) = \tan(x) \). The limit of the difference quotient is
We have already calculated some derivatives of elementary functions using the limit of the difference quotient of the function \(f(x)\).
On this page we will see how to calculate the derivative of the power function using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:
On this page we will see how to calculate the derivative of the exponential function using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:
The sign preservation theorem for sequences states that if a real sequence \( a_n \) converges to a limit \( L \neq 0 \), there exists an index \( N \) beyond which all terms of the sequence have the same sign as \( L \). In other words:
An equation is quadratic if and only if it can be written in the following form:
\[ a x ^ 2 + b x + c = 0 \quad , \quad a \neq 0 \]
The sign preservation theorem for functions states that if a real function \( f \) has a limit \( L \neq 0 \) as \( x \to x_0 \), then there exists a neighborhood of \( x_0 \) such that the function \( f(x) \) preserves the same sign as \( L \) for al
A first-degree equation is a first-degree polynomial set equal to zero. Generally, an equation is first-degree if it can be written in standard form:
\[ ax + b = 0 \quad \text{with} \quad a \neq 0 \]
Let \( a \neq 0 \) and let \( n \in \mathbb{N} \). The \( n \)-th power of \( a \), denoted by the symbol \( a^n \), is defined as the product of \( a \) by itself \( n \) times. Mathematically, this product is expressed as:
Even functions and odd functions are distinguished by their symmetries: even functions are symmetric with respect to the y-axis, while odd functions are symmetric with respect to the origin.
In this section, we will examine the steps for calculating the variance of a random variable that follows a Gamma distribution.
A function is a rule between two sets, which associates to each element of the first set (the domain) a unique element of the second set (the codomain).
A line is a primitive concept in Euclidean geometry, meaning it cannot be defined in more elementary terms but is taken as a fundamental entity.
An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points, called foci, is constant.
