Method

Numerical Methods for Equations

Approximate solutions, and sometimes exact solutions, to equations, can be found using numerical methods. These methods are useful for finding solutions to equations where algebraic methods either cannot by applied or are too time-consuming to be of practical use.

Method

Guess and Check

In the guess and check method, the idea is to first guess what the solution might be and then test the guess in the equation. The result of the test is then used to refine the guess, which is again tested in the equation. This is repeated until the guess is sufficiently close to the solution.

Solve $3^{x} = 17$ to two significant digits
$x$	$3^{x}$	Comment
$2$	$3^{2} = 9$	$3^{2} < 17 \Rightarrow x > 2$
$3$	$3^{3} = 27$	$3^{3} > 17 \Rightarrow x < 3$
$2.5$	$3^{2.5} \approx 15.58845 \dots$	$3^{2.5} < 17 \Rightarrow x > 2.5$
$2.6$	$3^{2.6} \approx 17.39863 \dots$	$3^{2.6} > 17 \Rightarrow x < 2.6$
$2.55$	$3^{2.55} \approx 16.46869 \dots$	$3^{2.55} < 17 \Rightarrow x > 2.55$

Since the solution to the equation is in the interval $2.55 < x < 2.6,$ an approximate answer with two significant digits is $x \approx 2.6 .$

Method

Graphing Method

In the graphing method, each side of the equation is interpreted as a function. These functions are then graphed on the same set of axes. The solutions to the equation are the $x$ -coordinates of the points of intersection between the graphs. In the following example, the solution to $x^{2} = 4^{x}$ is approximated.

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.

Method

Calculator

Many calculators have one or more tools for numerically approximating the solutions to equations. The steps to find $x$ for the equation $3^{x} - 6 = 0$ using a TI-84 calculator are the following. Press the button $MATH$ and look for the option Solver. The screen will then show this.

Next, the left-hand side of the equation should be written on the second line. Finally, by pressing the button $ENTER$ 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.

Method

Other Methods

Other commonly used numerical methods for solving equations include Newton's method, the bisection method, and the secant method. All of these methods use iterative approaches to finding the solution where each iteration yields a better approximation than the previous.

Method

Advantages and Disadvantages

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 are in most cases preferred.