An equation is a mathematical equality between two expressions \(F(x) = G(x)\) containing one or more variables, called unknowns. Equations are a fundamental mathematical tool used to express relationships between unknown variables and allow us to solve complex problems by determining the value of the unknowns they contain. Solving an equation means finding all its solutions in the definition set of the variables. Equations can be categorized based on their general form, which is represented by the expression:

\[F(x) = 0\]

Here is a summary of the different types of equations that are covered in Algebrica.

Algebraic equations

Algebraic equations are mathematical expressions that comprise only polynomials, which are functions consisting of a sum of terms, each term being the product of a constant and one or more variables raised to a non-negative integer power. These equations take the form of either \(P(x) = 0\) or \(P(x) = Q(x)\), where \(P(x)\) and \(Q(x)\) are two polynomials. For example:

\[P(x) = a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x+a_{0} = 0\]

Algebraic equations can be classified based on the degree of the monomial with the highest degree within the polynomial. Here are some common types of equations:

  • Linear equations have variables with a maximum exponent of \(1\). They take the form \(y = mx + b\) and represent a straight line in geometry.

  • Quadratic equations, with variables having a maximum exponent of \(2\), offer a variety of solutions. They take the form \(ax^2 + bx + c = 0\) and represent a parabola. They can have two distinct real solutions, single real or two complex solutions, adding to their intrigue.

  • Cubic equations have variables with a maximum exponent of \(3\). They take the form \(ax^3 + bx^2 + cx + d = 0\). These equations can have up to three real or complex solutions.

  • Equations of degrees higher than third involve variables with an exponent greater than \(3\). They can be fourth-degree, fifth-degree, sixth-degree, or even higher-degree equations.

Lastly, we’d like to mention the existence of binomial or trinomial equations, which are equations of a degree higher than the second. These types of equations can be solved by factoring them into first and second-degree polynomials or by using variable substitution methods to reduce them to a second-degree equation.

Rational equations

Rational equations are equations that contain at least one fraction where the numerator and denominator are both polynomials. In other words, a rational equation is a quotient of two polynomials, which can be expressed in the form of:

\[\frac{P(x)}{Q(x)} = 0\]

where \(Q(x)\) cannot be equal to zero.

Irrational equations

Irrational equations refer to equations where the variable is placed under a root sign. These equations are usually expressed as a root sign with a polynomial expression below it and they follow a specific form.

\[\sqrt[n]{x} = g(x)\]

with \(f(x)\) and \(g(x)\) two polynomials with real coefficients.

Absolute value equations

Absolute value equations involve using the absolute value function for the variable. They take the form \(|x| = a\) where \(x\) is the unknown and \(a\) a positive real number.

Transcendent equations

Transcendent equations are a class of equations where one or more variables are involved in transcendent functions, such as logarithmic, exponential, or trigonometric functions, that cannot be expressed through polynomial functions or algebraic equations.

  • Logarithmic equations are expressions involving a variable’s logarithmic function, such as \(2log_2(2-x)^2 = 0\). To solve these equations, one can apply various properties of logarithms, such as the change of base property or the logarithmic equivalence property. One can manipulate the equation to isolate the variable and obtain its solution using these properties.

  • Exponential equations are a class of equations in which the variable appears as an exponent of an exponential base. These equations involve expressions where the variable is raised to a power, such as \(a^x = b\), where \(a\) and \(b\) are constants and \(x\) is the unknown.

  • Trigonometric equations involve periodic trigonometric functions such as \(sin(x)\), \(cos(x)\), or \(tan(x)\) with a variable. They can have infinite solutions.