Many of the algebraic operations may be performed on a graphing calculator. Here we explain how to use some of the options listed in the ‘Algebra’ Menu of the Home Screen of the TI-92 Graphing Calculator. When using new operations try them on problems to which you know the answer. Also try to estimate the results so that you have an idea if you pressed the correct keys.

When starting you should clear the home screen and the entry line:    8  then display the ‘Algebra’ Menu: 

For the zeros of a function, use item 4:zeros in the Algebra Menu, 

Type the rule of correspondence followed by a comma, the variable, and right parentheses. Then press .

Example. Find the zeros of f (x) = 2x² - 98.
Solution:  4 2x  2  98  x  

The answer displayed is: (- 7 7)

Problem. Find the zeros of f (x) = 3x² - 75.

Example. Find the zeros of f (x) = x² - 5x - 36.
Solution:  4 x  2  5x  36  x  

The answer displayed is: (- 4 9 )

Problem. Find the zeros of f (x) = x² + 3x - 28.

Example. Find the zeros of f (x) = 3x² + 12.
Solution:  4 3x  2  12  x  

The answer displayed is: ( ) (which indicates no real zeros)

Looking at the associated equation we suspect complex or imaginary zeros (for 3x² +12 = 0, x² = - 4). The graphing calculator can handle this. We select item A: Complex in the Algebra menu and cZeros in the sub-menu.

A 3 3x  2  12  x  

The answer displayed is: ( - 2i 2i )

Problem. Find the zeros of f (x) = 2x² + 18.

Example. Find the zeros of f (x) = 3x² - x + 4.
Solution:  4 3x  2  x  4  x   displays ( )

Try using cZeros:  A 3 3x  2  x  4  x  

The answer displayed is: 

For decimal form:   displays: (.166667 - 1.14261 i .166667 + 1.14261 i )

When the answer is not completely displayed (a right pointing triangle at the right end), highlight the answer by pressing . Then press  several times until the answer is completely displayed.  will remove the highlight.

Problem. Find the zeros of f (x) = 3x² - 2x + 7.

In the Algebra Menu, Item A:Complex and cZeros in the sub-menu will give real zeros as well as complex zeros. Do the above example using:  A 3 and so forth. Did you get the same results?

To solve an equation use item 1:solve in the Algebra Menu .

Type the equation followed by a comma, the variable, and right parentheses. Then press .

Example. Solve x² - 5x - 36 = 0.
Solution:  1 x  2  5x  36  0  x  

The answer displayed is: x = 9 or x = - 4

Problem. Solve x² + 3x - 28 = 0.

Example. Solve 3x² - x + 4 = 0.
Solution:  1 3x 2  x  4  0  x  

The answer displayed is: false

Try using cSolve:  A1 3x 2  x  4  0  x  

The answer displayed is: 

For decimal form:   displays: (.166667 - 1.14261 i .166667 + 1.14261 i )

In the Algebra Menu, Item A:Complex and cSolve in the sub-menu will give real roots as well as complex roots. Do the above example using  A1 and so forth. Did you get the same results?

To factor an expression, use item 2:factor in the Algebra Menu, .

Type the expression and right parentheses. Then press .

Example. Factor x² - 9.
Solution:  2 x 2  9  

The answer displayed is: (x - 3)× (x + 3)

Problem. Factor (a) 4x² - 25. (b) x² - 7. (Does   help?)

Example. Factor x² + 2ax + a².
Solution:  2 x 2  2 a  x  a  2  

The answer displayed is: (x + a)²

Note that when entering the product of literal numbers the multiplication sign  must be used. Try entering the product of 2ax as 2ax. Did you get the correct answer?

Problem. Factor x² - 2ax + a².

Example. Factor x² + 4.
Solution:  2 x 2  4  

The answer displayed is: x² + 4 (the expression we were trying to factor). Try   . Again the display is: x² + 4. This implies that the factors may be complex. This is handled by selecting item A: Complex in the Algebra menu. Then cFactor in the sub-menu.

A 2 x 2  4  

The answer displayed is: (x + -2 i)× (x + 2 i)

Which is the same as (x - 2i)(x + 2i).

Problem. Factor 9x² + 25.

In the Algebra Menu, Item A:Complex and cFactor in the sub-menu will give real factors as well as complex.

Example. Factor x² + x + 1.
Solution:  A2 x 2  x  1  

The answer displayed is: x² + x + 1

Now press  

The answer now displayed is: (x + .5 - .866025× i )× (x + .5 + .866025× i)

Problem. Factor 3x² + 2x + 1.

To expand an expression, use item 3:expand in the Algebra Menu, .

Type the expression followed by a right parentheses. Then press .

Example. Expand (x + a)².
Solution:  3  x  a   2  .

The answer displayed is: x² + 2ax + a²

Problem. Expand (x - a)².

Example. Expand (x + a)(x - a).
Solution:  3  x  a   x  a   .

The answer displayed is: x² - a²

Problem. Expand (x - 3)(x + 3).

Example. Expand (2x - 1)(3x + 4) = 0.
Solution:  3  2 x  1   3 x  4   0  

The answer displayed is: 6x² + 5x - 4 = 0

Problem. Expand (5x + 1)(- 2x + 3) = 0.

When operating on complex numbers the imaginary number i is obtained by:   (above the I key). Pressing    2  displays - 1 and    3  displays - i ; so i2 = -1 and i3 = - i. What is i4 ?

Example. Expand (a + bi)(a - bi).
Solution:  3  a  b     a  b     

The answer displayed is: a² + b²

Problem. Expand (a) (3 - 2i)(3 + 2i), (b) (3 + 2i)(2 - 3i).

Problem. Solve the system.

Solution: Display and clear the home screen:    8 Clear the entry line: 

Solve the first equation for the linear variable (here y):

1 (Solve) x  2  y  5  y  

The solution is in the lower right of the history area: y = - x² + 5

Solve the second equation for x with the substitution of the y solution of the first equation:

1 x  y  1  x      (pastes the solution for y on the entry line) 

The display shows: x = 2 or x = - 3

We now have two values of x to use in finding y. We use one at a time.

    (pastes the solution for y on the entry line)

  x  2  displays y = 1

   3  displays y = - 4

Thus the solution set is {(- 3, - 4), (2, 1)}.

Problem. Solve the system.

Example. Use a graphing calculator to shade the area bounded below by y = x - 1 and above by y = - x² + 5.
Solution: Clear the home screen, Y editor screen, and entry line:

   8    8  

Graph the functions:    x  2  5  x  1   6

From the ‘Math’ menu select ‘Shade’:  C Answer question in lower left corner of screen.

Above ?: Use  or  to select the function shading will be above. Press 

Below ?: Use  or  to select the function shading will be below. Press 

Lower Bound ?: Type the x value, here - 3, (or use  or  to move the cursor to the lower bound) Press 

Upper Bound ?: Type the x value, here 2, (or use  or  to move the cursor to the upper bound) Press 

The required area is shaded.

Problem. Use a graphing calculator to shade the area bounded above by y = - x - 1 and below by y = x² - 3.
 

To Table of Contents

To Section 6 & 7. Solutions of Linear Systems

To Section 9. Operations with polynomials