Equation

In mathematics, equations are equalities between monomials or polynomials, for which the purpose is to search for the numerical value of one or more literal variables, called unknown (for example \(x\)), which make the equality true. This value is called the solution or root of the equation.

An equation is written as two expressions, connected by an equals sign (“=”) \(x+1=0\). The expressions on the two sides of the equals sign are called the “left-hand side” and “right-hand side” of the equation. The equations can be:

  • whole: if the unknown is present only in the numerator;
  • rational: if the unknown is also present in the denominator;
  • determined: if they have finite number solutions;
  • indeterminate: if they have infinite solutions;
  • impossible: if they have no solution.

Quadratic equation

A quadratic equation (from the Latin quadratus for “square”) is any equation that can be rearranged in standard form as:

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

where \(x\) represents an unknown, and \(a\), \(b\), and \(c\) represent known numbers, where \(a\neq 0\). If \(a=0\), then the equation is linear, not quadratic, as there is no \(ax^2\) term. The numbers \(a\), \(b\), and \(c\) are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. From which two solutions of the equation are obtained by applying the following formula:

\[x_{1,2}=\dfrac{-b\pm\sqrt{b^2+2ac}}{2a}\]

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