Finding Zeros, Roots, Solutions, x-intercepts

Zeros or roots of functions and solutions to equations can either be real or complex numbers. x-intercepts can only be real numbers.

The TI graphing calculator can be used to find the real roots of an equation.

All calculators

  1. If your equation has something on the right side, subtract it from both sides so that the right hand side is 0.
  2. Go to Y=
  3. Enter the expression
  4. Hit Graph
  5. Change the viewing window if necessary so that you can see where the graph crosses the x-axis. You need to make sure that there is enough to the left and right of the x-intercept to select a point.
  6. Continue on to the specific steps for your calculator. If there is more than one x-intercept, you may need to change your viewing window and repeat the steps for each x-intercept.

TI-82

  1. Press Calc (2nd Trace)
  2. Choose Root (#2)
  3. For the Lower Bound, arrow to the left of the x-intercept and press enter.
  4. For the Upper Bound, arrow to the right of the x-intercept and press enter.
  5. For the Guess, arrow to the x-intercept and press enter.
  6. The TI-82 will return a value for x and 0 (or really close to it) for y.

TI-83

For the TI-83, you may enter left bounds, right bounds, and guesses from the keyboard instead of arrowing and hitting enter. When it asks for a value, just enter your number and press enter.

  1. Press Calc (2nd Trace)
  2. Choose Zero (#2)
  3. For the Left Bound, arrow to the left of the x-intercept and press enter.
  4. For the Right Bound, arrow to the right of the x-intercept and press enter.
  5. For the Guess, arrow to the x-intercept and press enter.
  6. The TI-83 will return a value for x and 0 (or really close to it) for y.

TI-85 / TI-86

  1. Press Graph
  2. Press More and then Math (F1)
  3. Press Lower (F1). Arrow to the left of the x-intercept and press enter.
  4. Press Upper (F2). Arrow to the right of the x-intercept and press enter.
  5. Press Root (F3). Arrow to the x-intercept and press enter.
  6. The TI-85 will return a value for x and 0 (or really close to it) for y.

TI-89 / TI-92

For the TI-89 and TI-92, you may enter lower bounds or upper bounds from the keyboard instead of arrowing and hitting enter. When it asks for a value, just enter your number and press enter.

  1. Press Math (F5)
  2. Choose Zero (#2)
  3. For the Lower Bound, arrow to the left of the x-intercept and press enter.
  4. For the Upper Bound, arrow to the right of the x-intercept and press enter.
  5. The TI-89 will return a value for x and 0 for y.

You do not need to supply a guess for the TI-89. You should make sure there is only one x-intercept in the interval specified or you won't know which one it is going to give you.

Example (TI-83+)

Find all real solutions to the equation exp(-x2) = -x.

Note: exp(x) is the exponential function and is often written as ex. It is ex (2nd LN) on the calculator, but gets displayed as e^(x).

Begin by moving the -x to the left side where it becomes +x. The function to graph is y1 = x + e^(-x2).

Graph Zero Left Right Guess Solution
Initial graph Choose Zero from calc menu Enter left bound Enter right bound Enter guess Calculator gives solution

The calculator says the solution is x = -0.6529186. The y-value of 0 is ignored. There was no y in the original problem.