Logic Equations

Abstract

For an easy access to these concepts we will go back to the basic understanding of an equation, as it is known from elementary mathematics. An equation ax+ b = c or ax ² + bx + c = d is a constraint for the values of x, and a solution x ₁ allows the transformation of the equations in the identities c = c by calculating ax ₁ + b which results in c or by calculating ax ₁ ² + bx ₁ + c that has to result in d. It is well known that these equations are equivalent to the equations ax + (b - c) = 0 or ax ² + bx + (c - d) = 0 (i.e. these equations have the same solutions as the original equations) . Further transformations result in $x + \frac{{b - c}}{a} = 0$ or ${x^2} + \frac{b}{a}x + \frac{{c - d}}{a} = 0$ which can be changed to x + x ₀ = 0 or x ² + px +q = 0, and the solutions are

$$x = - {x_0}\quad {x_{1,2}} = - \frac{p}{2} \pm \sqrt {\frac{{{p^2}}}{4} - q,} $$

, under consideration of several conditions, like $a \ne 0,\frac{{{p^2}}}{4} - q \geqslant 0$.