Still the Ancient Greek mathematician Diophantus Aleksandriysky for the indication of unknown number entered alphabetic references. The most frequent among unknown - X, we put it by default, every time working out the equation or inequality. Though we can use any other not digital symbol. The equations in which except numbers only one unknown – X, and ways of their decision, we now will also consider.
Instruction
1. To solve the equation - means, to find all its roots. The equation root, that is value of the unknown at which the equation becomes true can be one or not. Korney can be several, infinite number or not to be at all.
2. Matters at the solution of the equation a function range of definition. The fact is that at some values x the equation loses the meaning. So, for example, the denominator cannot be equal to zero so if in the equation there are fractions with participation x in a denominator, then the area of permissible values is limited. The first step at the solution of any equation is to define its area of permissible values. You remember: the root of even degree can have no negative radicand, the denominator cannot be equal to zero, trigonometrical functions have own restrictions, etc.
3. In the course of the solution of the equation, we simplify it, gradually reducing to easier for us, but with the same roots to the equation. We can transfer the composed equations on the one side of the sign equally to another, changing a minus sign to plus and vice versa. We can increase both members of equation, divide or change somehow differently, but it is surely symmetric, that is is identical both the right, and left members of equation. We can remove the brackets and take out for them. To perform the arithmetic operations specified in the equation according to rules. Actually in it process of the decision also consists. To lead the equation to a "decent" look and then to learn its roots.
4. The first in a school course the linear equations from one unknown are considered. These equations have generally an appearance: ax+b=0. Here an and b of designation for numerical values. The solution of the equation looks so: x=-b/a. Having received difficult looking equation for the decision, we try in to give it a habitual type of linear. For what if in the equation there are fractional expressions, we reduce all composed equations to a common denominator. Then we multiply both members of equation by this denominator. We remove all brackets. We transfer everything composed including x on one party of the equation. All without unknown on opposite. We put, we will read, we perform all required and possible operations. Which usually lead us to the fact that from each party from the sign equally there is only one composed. It was necessary only to divide composed without x, into coefficient near the unknown.
5. It is convenient to solve many equations graphically. For this purpose all of us composed collect on the one side of the equation. On the other hand zero is formed. Replace it with y, draw coordinate axes and construct the schedule, the function which is available now in existence. Places of crossing of the schedule with abscissa axis are roots. Write down.
6. When you found out all roots of the equation, do not forget to compare results from the function found earlier a range of definition. Out of its limits there are no roots because also the equation does not exist.