Functions: \(f\), \(g\), \(h\)

Unknowns (independent variables): \(x,\) \(y,\) \(z\)

Solutions (roots) of equations: \(x,\) \({x_1},\) \({x_2},\) \({y_1},\) \({y_2},\) \({y_3}\)

Unknowns (independent variables): \(x,\) \(y,\) \(z\)

Solutions (roots) of equations: \(x,\) \({x_1},\) \({x_2},\) \({y_1},\) \({y_2},\) \({y_3}\)

Real numbers: \(a\), \(b\), \(c\), \(d\), \(p\), \(q\), \(u\), \(v\)

Discriminant: \(D\)

Discriminant: \(D\)

- An equation is an equality of the form

\(f\left( {x,y,z, \ldots } \right) =\) \( g\left( {x,y,z, \ldots } \right),\)

where \(f\) and \(g\) are functions of one or several unknowns (independent variables). - The solutions (roots) of an equation are the values of the unknowns which satisfy the equation, i.e. make the equality true.
- Two equations are said to be equivalent if they have the same set of solutions.
- Moving any term of an equation from one side to another and changing its sign produces an equivalent equation.

\(f\left( x \right) + h\left( x \right) = g\left( x \right)\; \Leftrightarrow\) \(f\left( x \right) =\) \( g\left( x \right) – h\left( x \right)\) - Adding the same expression or quantity to the both sides of an equation produces an equivalent equation.

\(f\left( x \right) = g\left( x \right)\; \Leftrightarrow\) \(f\left( x \right) + h\left( x \right) =\) \( g\left( x \right) + h\left( x \right)\)

Note: The equivalence of equations is violated if the function \(h\left( x \right)\) is not defined for values of \(x,\) which are solutions of the equation. - Subtracting the same expression or quantity from each side of an equation produces an equivalent equation.

\(f\left( x \right) = g\left( x \right)\; \Leftrightarrow\) \(f\left( x \right) – h\left( x \right) =\) \( g\left( x \right) – h\left( x \right)\)

Note: The equivalence of equations is violated if the function \(h\left( x \right)\) is not defined for values of \(x,\) which are solutions of the equation. - Multiplying both sides of an equation by the same nonzero number produces an equivalent equation.

\(f\left( x \right) = g\left( x \right)\; \Leftrightarrow\) \(f\left( x \right) \cdot c = g\left( x \right) \cdot c\;\) \(\left( {c \ne 0} \right)\) - Dividing both sides of an equation by the same nonzero number produces an equivalent equation.

\(f\left( x \right) = g\left( x \right)\; \Leftrightarrow\) \(f\left( x \right)/c = g\left( x \right)/c\;\) \(\left( {c \ne 0} \right)\) - Multiplying both sides of an equation by an expression involving variables can introduce extraneous roots or cause the loss of roots, i.e. can yield non-equivalent equations.
- Squaring or raising both sides of an equation to an even power can introduce extraneous roots.
- The product is equal to zero if any of the factors is zero.

\(f\left( x \right) \cdot g\left( x \right) = 0\; \Leftrightarrow\) \(f\left( x \right) = 0 \cap g\left( x \right) = 0\) - Linear equation with one variable \(ax + b = 0\)
- Solution of a linear equation with one variable

\(ax + b = 0,\;\; \Rightarrow\) \(x = – {\large\frac{b}{a}\normalsize}\;\;\left( {a \ne 0} \right)\) - Linear equation with two variables \(ax + by + c = 0\)
- Quadratic equation \(a{x^2} + bx + c = 0\)
- Discriminant of a quadratic equation \(D = {b^2} – 4ac\)
- Roots of a quadratic equation

\(a{x^2} + bx + c = 0,\;\;\) \({x_{1,2}} = {\large\frac{{ – b \pm \sqrt D }}{{2a}}\normalsize} =\) \( {\large\frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\normalsize}\) - Reduced quadratic equation and its solutions

\({x^2} + px + q = 0,\;\;\) \({x_{1,2}} =\) \( – {\large\frac{p}{2}\normalsize} \pm \sqrt {{{\left( {{\large\frac{p}{2}}\normalsize} \right)}^2} – q} \) - Vieta’s formulas

\({x^2} + px + q = 0,\; \Leftrightarrow\) \({x_1} + {x_2} = – p,\;\) \({x_1}{x_2} = q\) - Incomplete quadratic equation \(\left( {c = 0} \right)\)

\(a{x^2} + bx = 0,\;\) \({x_1} = 0,\;\) \({x_2} = – b/c\) - Incomplete quadratic equation \(\left( {b = 0} \right)\)

\(a{x^2} + c = 0,\;\) \({x_{1,2}} = \pm \sqrt { – {\large\frac{c}{a}}\normalsize} \) - Incomplete quadratic equation \(\left( {b = c = 0} \right)\)

\(a{x^2} = 0,\;\) \({x_1} = {x_2} = 0\) - Cubic equation (in a canonical form) \(a{x^3} + b{x^2} +\) \( cx + d = 0\)
- Reduced cubic equation \({y^3} + py + q = 0\)

Transformation of a cubic equation from the canonical form to the reduced form can be done by using the substitution \(x = y – b/\left( {3a} \right).\) - Cardano’s formula

\({y^3} + py + q = 0,\)

\({y_1} = u + v,\;\) \({y_{2,3}} = – {\large\frac{1}{2}\normalsize}\left( {u + v} \right) \pm\) \( {\large\frac{{\sqrt 3 }}{2}\normalsize}\left( {u – v} \right)i,\) where

\({i^2} = – 1,\;\) \(u =\) \(\sqrt[\large 3\normalsize]{{ – {\large\frac{q}{2}\normalsize} + \sqrt {{{\left( {{\large\frac{q}{2}}\normalsize} \right)}^2} + {{\left( {{\large\frac{p}{3}}\normalsize} \right)}^2}} }},\;\) \(v =\) \(\sqrt[\large 3\normalsize]{{ – {\large\frac{q}{2}\normalsize} – \sqrt {{{\left( {{\large\frac{q}{2}}\normalsize} \right)}^2} + {{\left( {{\large\frac{p}{3}}\normalsize} \right)}^2}} }}.\)