# Equations

Functions: $$f$$, $$g$$, $$h$$
Unknowns (independent variables): $$x,$$ $$y,$$ $$z$$
Solutions (roots) of equations: $$x,$$ $${x_1},$$ $${x_2},$$ $${y_1},$$ $${y_2},$$ $${y_3}$$
Real numbers: $$a$$, $$b$$, $$c$$, $$d$$, $$p$$, $$q$$, $$u$$, $$v$$
Discriminant: $$D$$
1. 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).
2. The solutions (roots) of an equation are the values of the unknowns which satisfy the equation, i.e. make the equality true.
3. Two equations are said to be equivalent if they have the same set of solutions.
4. Moving any term of an equation from one side to another and changing its sign produces an equivalent equation.
$$f\left( x \right) + h\left( x \right) = g\left( x \right)\; \Leftrightarrow$$ $$f\left( x \right) =$$ $$g\left( x \right) – h\left( x \right)$$
5. 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.
6. 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.
7. Multiplying both sides of an equation by the same nonzero number produces an equivalent equation.
$$f\left( x \right) = g\left( x \right)\; \Leftrightarrow$$ $$f\left( x \right) \cdot c = g\left( x \right) \cdot c\;$$ $$\left( {c \ne 0} \right)$$
8. Dividing both sides of an equation by the same nonzero number produces an equivalent equation.
$$f\left( x \right) = g\left( x \right)\; \Leftrightarrow$$ $$f\left( x \right)/c = g\left( x \right)/c\;$$ $$\left( {c \ne 0} \right)$$
9. 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.
10. Squaring or raising both sides of an equation to an even power can introduce extraneous roots.
11. The product is equal to zero if any of the factors is zero.
$$f\left( x \right) \cdot g\left( x \right) = 0\; \Leftrightarrow$$ $$f\left( x \right) = 0 \cup g\left( x \right) = 0$$
12. Linear equation with one variable $$ax + b = 0$$
13. Solution of a linear equation with one variable
$$ax + b = 0,\;\; \Rightarrow$$ $$x = – {\large\frac{b}{a}\normalsize}\;\;\left( {a \ne 0} \right)$$
14. Linear equation with two variables $$ax + by + c = 0$$
15. Quadratic equation $$a{x^2} + bx + c = 0$$
16. Discriminant of a quadratic equation $$D = {b^2} – 4ac$$
17. Roots of a quadratic equation
$$a{x^2} + bx + c = 0,\;\;$$ $${x_{1,2}} = {\large\frac{{ – b \pm \sqrt D }}{{2a}}\normalsize} =$$ $${\large\frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\normalsize}$$
18. Reduced quadratic equation and its solutions
$${x^2} + px + q = 0,\;\;$$ $${x_{1,2}} =$$ $$– {\large\frac{p}{2}\normalsize} \pm \sqrt {{{\left( {{\large\frac{p}{2}}\normalsize} \right)}^2} – q}$$
19. Vieta’s formulas
$${x^2} + px + q = 0,\; \Leftrightarrow$$ $${x_1} + {x_2} = – p,\;$$ $${x_1}{x_2} = q$$
20. Incomplete quadratic equation $$\left( {c = 0} \right)$$
$$a{x^2} + bx = 0,\;$$ $${x_1} = 0,\;$$ $${x_2} = – b/c$$
21. Incomplete quadratic equation $$\left( {b = 0} \right)$$
$$a{x^2} + c = 0,\;$$ $${x_{1,2}} = \pm \sqrt { – {\large\frac{c}{a}}\normalsize}$$
22. Incomplete quadratic equation $$\left( {b = c = 0} \right)$$
$$a{x^2} = 0,\;$$ $${x_1} = {x_2} = 0$$
23. Cubic equation (in a canonical form) $$a{x^3} + b{x^2} +$$ $$cx + d = 0$$
24. 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 $$x = y – b/\left( {3a} \right).$$
25. 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
$${i^2} = – 1,\;$$ $$u =$$ $$\sqrt[\large 3\normalsize]{{ – {\large\frac{q}{2}\normalsize} + \sqrt {{{\left( {{\large\frac{q}{2}}\normalsize} \right)}^2} + {{\left( {{\large\frac{p}{3}}\normalsize} \right)}^2}} }},\;$$ $$v =$$ $$\sqrt[\large 3\normalsize]{{ – {\large\frac{q}{2}\normalsize} – \sqrt {{{\left( {{\large\frac{q}{2}}\normalsize} \right)}^2} + {{\left( {{\large\frac{p}{3}}\normalsize} \right)}^2}} }}.$$