Quadratic equations are an essential mathematical concept extensively employed in several fields, including engineering, science, physics and economics. A quadratic equation is a second-degree polynomial equation in one variable. The standard form of a quadratic equation presents a precise and organized arrangement of its terms. It is commonly expressed in the form of:

\[ax^2+bx+c=0 \]

where \(a\), \(b\), and \(c\) represent coefficients, and \(x\) is the variable. In mathematical terms, the degree of the equation is determined by the highest power of the variable. In this case, the highest power is two, also known as the quadratic term.

  • The coefficients \(a\), \(b\), and \(c\) are constants
  • \(x\) represents the variable.
  • \(a\) is the coefficient of the quadratic term \(x^2\), \(b\) the coefficient of the linear term \(x\) and \(c\) the constant term.
  • When \(a\) is equal to zero, the equation degenerates into a linear equation \(bx + c = 0\), or a constant equation, depending on the values of \(b\) and \(c\).

Quadratic equations are a particular type of trinomial equation with the exponent \( n \) equal to 1:

\[ ax^{2n} + bx^{2} + c = 0 \quad \text{with} \quad n = 1 \]


Resolution methods

A quadratic equation is incomplete when either the coefficient \(b\) or \(c\) is equal to zero. When this occurs, the solution of the equation can be derived immediately without the need for complex calculation

The most practical way of solving this type of equation is to use the quadratic formula, while in some cases, using the factorisation method is more immediate and straightforward.

The first step to solving a quadratic equation is generally to bring all terms into the standard form \(ax^2+bx+c=0\) to simplify the resolution. This form is handy for identifying the equation’s coefficients and determining the discriminant \(\Delta = b^2-4ac\) which helps determine the solutions to the equation.

  • The solutions to a quadratic equation are the values of \(x\) that satisfy the equation.
  • The fundamental theorem of algebra guarantees that a quadratic equation always has at most two solutions if considering \(x_1, x_2\), if a real double root is counted for two \(x=x_1, x=x_2\), and complex roots are included.

Quadratic formula

Given a quadratic equation in the standard form \(ax^2+bx+c = 0\), the quadratic formula is:

\[ x_{1,2} = \frac{{-b \pm \sqrt{{b^2 – 4ac}}}}{{2a}} \]

  • \(a, b, c \;\) are the coefficients of the equations and \(a \neq 0\).
  • The plus-minus symbol indicates that the quadratic equation has two solutions.
  • A quadratic equation always has at most two solutions if a real double root is counted for two \((x = x_1, x = x_2)\), and complex roots are included.

In a second-degree equation with a non-negative discriminant, the sum of the roots is given by the ratio \(\frac{-b}{a}\) and their product by \( \frac{c}{a} \).


Factoring

A quadratic equation can be factored into an equivalent equation of the form:

\[ax^{2}+bx+c=0 \to (x-x_1)(x-x_2)=0\]

where \(x_1\) and \(x_2\) are the roots of the equation. In general, we must find two numbers \(x_1, x_2\) that meet the following constraints \(x_1 \cdot x_2=a \cdot c \;\) and \(x_1+x_2=b \).

This method is straightforward for relatively simple polynomials, in which the solutions of the associated equation can be conveniently found, while it is impractical in more complex cases, in which it is preferable to use the quadratic formula.