Understanding the Ti 84 Solve Function

This article explains how the Ti 84 solve function can help obtain solutions to algebraic equations. For best understanding you will want to have a Ti 83 or Ti 84 calculator along with its guidebook and have passed or be taking Algebra I.

To work with the Ti 84 solve function, begin by keying Math, then 0. This should bring up a screen that says:

EQUATION SOLVER
Eqn: 0=

If "EQUATION SOLVER" doesn't appear on the top line, use the up-arrow key to move the cursor to the top after which "EQUATION SOLVER" should appear on the top line. If anything appears after 0=, use the CLEAR key to remove it.

To solve an equation such as X-2= 2X+4, begin by rewriting it so that the left side is zero. In this case, subtract X-2 from both sides, and enter the result after the = sign. This may be expressed as either 2X+4-(X-2), or simply X+6. Then ENTER, and see:


2X+4-(X-2) = 0
_X=?
bound = {? (Disregard for now or see manual)
_left-rt=? (Disregard for now or see manual)

The “?” means that the actual value, at this point, is unpredictable and irrelevant.

Now move the cursor to the X=? line and ALPHA/SOLVE (ALPHA first).The screen will change to:


2X+4-(X-2) = 0
_X=-6
The calculated value of X.
bound= {? (Disregard for now or see manual)
_left-rt=0 (Disregard for now or see manual)


The Ti 84 solve function determines the value of X by a numerical-approximation technique. Beginning with a "guess" (the initial value on the X= line), it evaluates the top-line equation. If this is non-zero, it tries a different value, and determines whether the result is closer to, or farther away from zero. Based on the result, it changes the value and tries again continuing until the result is either zero or it is impossible to make it any closer to zero. There are two reasons why the latter may occur.

Starting with X^2-2=0, it will soon determine that X is between 1.41 and 1.42. Making smaller and smaller changes, it will soon find a value which is accurate to 14 significant digits (the most available). Since the square root of 2 is an irrational number, no decimal value of X^2-2 can equal zero; but the value of x will be as accurate as can be represented with the available digits.

Starting with X^2+2=0, it will soon determine that no value of X will reduce the expression to less than +2; "ERR: NO SIGN CHNG" will be returned. The Ti 84 solve function operates only on real numbers. (Other Ti 84 functions are able to handle imaginary numbers, but the SOLVE function is not).

A more sophisticated example would involve S=V^2/ (2A); stopping distance. When entered this might appear as:

S-V^2/ (2A) =0
_S=??? (Distance to be determined)
_V=64 (Initial speed 64 ft/sec)
_A=8 (Braking; 8 ft/sec^2)
bound= {? (Disregard for now or see manual)
_left-rt=? (Disregard for now or see manual)


Note the parentheses around 2A. These are critical to have the A interpreted as part of the denominator. V^2/2A would be interpreted as (V^2/2)*A; incorrect. For similar reasons, 1/ (2+x) must be written as such. The Ti 84's Order of Operations, although fairly standard, may not be documented in shorter versions of the manual; when in doubt, use parentheses. This equation can be solved for any one variable, as long as the other two are specified. Place the cursor on the variable to be solved for then ALPHA/SOLVE, if the variable can be calculated it will be displayed under a flashing cursor.

This approach can be used in cases where an algebraic solution cannot be obtained. P=N*sin (180/N) gives the perimeter of an N-sided regular polygon (of unit radius). This cannot be solved algebraically for N. But the Ti 84 solve function, using the trial-and-error method, quickly returns a value. When an equation has more than one real root, the Ti 84 solve function tends to return the one which is closest to the initial "guess". As a practical matter, it is advisable to know how many real roots exist.

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