How to solve problems with parameters

If the given equation or inequality can be simplified, be sure to use it. Apply standard methods for solving equations, as if the argument were a normal number. As a result, you will be able to Express the variable via a parameter, for example, x=R/2. If the solution of equation you met with no restrictions on the parameter value (it is not necessary under the root sign, under the sign of the logarithm in the denominator), record the answer, putting that he found for all valid values of the parameter R.

For solving problems with standard charts (e.g., line, parabola, hyperbola) use the graphical method. Divide the area of parameter values for such intervals where the value of the variable (or variables) will be different, and for each interval, draw a line graph. Pay special attention to the extreme points of the lines to precisely determine their belonging to the schedule, substitute this value into the function and solve the equation with him. If the equation at this point has no solution (for example, is obtained by dividing by zero), eliminate it from the graph, noting an empty circle.

To solve the problem with respect to the parameter, pick the variable and the argument for equal members of equations or inequalities and simplify the expression. Then return to the original sense of the members and consider the solution of the problem for all possible parameter values. For this set of parameter values you need to divide into intervals.

When you search for the boundaries of intervals, pay attention to those expressions involving a parameter. For example, you have the expression (a-5), among the borders of intervals have to be number 5, since this value reverses the value in the parentheses to 0. Great importance is the expression under the sign of the parameter of division, root, module , etc.

When you find the limits of intervals, consider its function for each of them. To simplify this, just substitute into the function one of the numbers from this interval and solve the resulting problem. Often, just by substituting different values, it is possible to find the right way of solving the problem.