Approximate , and sometimes exact solutions to , can be found using numerical methods. These methods are useful for finding solutions to equations where algebraic methods either cannot be applied or are too time-consuming to be of practical use.
In the guess and check method, the idea is to first guess what the solution might be followed by testing that guess in the equation. The test result is then used to refine the guess, which is again tested in the equation. This process is repeated until the test result from the guess is sufficiently close to the solution.
| Solve 3x=17 to two significant digits
Since the equation's solution is in the interval 2.55<x<2.6, an approximate answer with two is x≈2.6.
In the graphing method, each side of the equation is interpreted as a . These functions are then graphed on the same set of axes. The solutions to the equation are the x
-coordinates of the between the graphs. For example, to approximate the solution of the equation x2=4x,
consider the following functions.
Then, both functions are graphed on the same coordinate plane.
The solution is in the interval -0.75<x<-0.5. To find a better approximation it is necessary to readjust the scale on the x-axis and to narrow the interval on which the functions are graphed.
Many calculators have one or more tools for numerically approximating the solutions to equations. These are the steps to find x
for the equation 3x−6=0
using a TI-84 calculator. First, press the button MATH
and look for the option
. The screen will then show this.
Next, the left-hand side of the equation should be written on the second line. Then, by pressing the ENTER button, a numerical approximation of the solution is calculated.
To use this particular tool, one side of the equation needs to be equal to 0. Therefore, it may be necessary to rearrange the equation before writing it in the calculator.
Other commonly used numerical methods for solving equations include Newton's method, the bisection method, and the secant method. These methods use iterative approaches to finding the solution where each iteration yields a better approximation than the previous.
Numerical methods are helpful when solving complicated equations that cannot be solved algebraically. However, when using numerical methods it is often only possible to find approximations of the solutions. If it is necessary to find an exact solution, algebraic methods in most cases are preferred.