HP 33S User manual

hp calculators
HP 33S Solving for roots
Roots of an equation
Using the SOLVE function
Practice solving problems involving roots
Image of the cursor key area of the HP 33S calculator

hp calculators
HP 33S Solving for roots
Roots of an equation
The roots of an equation are values of X where the value of Y is equal to zero. For example, the equation Y = X – 2 has
a real root at the value +2. The equation Y = X2– 9 has real roots at the values of + 3 and – 3. Not every equation has
roots that are real numbers. For example, the equation Y = X2+ 4 has no real roots, meaning there are no real values for
X that will cause X2+ 4 to equal zero.
Using the SOLVE function
The HP 33S has a very powerful root finding capability built into its SOLVE function. As applied in this training aid, the
SOLVE function, accessed by pressing the Ûkey, will be used to find roots from user-written programs computing
the value of a function. This will involve entering a small program, keying in a small equation into the program using a
variable, indicating to the HP 33S which variable is being considered as the current function, and then solving for the
value of that variable when the function is equal to zero. The HP 33S knows which variable to solve for by setting the
value of the function under consideration using the ºs function. To indicate to the HP 33S that the variable X is to
be used, press ºsX.
This training aid cannot begin toillustrate the wide range of applications available using the built-in solver, but it can
illustrate some of themore common uses. For additional information, see chapters 7 and 14 of the HP 33S RPN/ALG
Scientific Calculator Owner’s Manual.
Practice solving problems involving roots
Example 1: Solve for the roots of Y = X2– 4
Solution: We're looking for values of X such that X2– 4 = 0. First, we'll enter a program that computes the value of
the function. If a program already exists in program memory with the name of X, then it will need to be
cleared. This can be done by pressing ¹uÕÏ to have the HP 33S display the list of
programs in the calculator and then press Øto step through the program labels. When the label of the
program to be deleted is shown in the display, pressing ¹¡will delete that program from the
calculator's memory. Pressing Åwill then clear the display and allow you to proceed.
In RPN or algebraic mode: ¹£¹ÓXºdhX)2Ã4Ï
ºÔ
Figure 1
To show the checksum and length of this program, press the following in RPN or algebraic mode. Note that
the symbol Õ means to press the right arrow direction of the cursor key at the top of the HP 33S. If you
are unsure of its location, see the image at the end of this training aid.
In RPN or algebraic mode: ¹uÕϺÎ
Figure 2
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hp calculators
HP 33S Solving for roots
If the checksum of the program just entered does not equal 9EAF, then you have not entered it correctly.
To clear the checksum display press:
In RPN or algebraic mode: ¹£
Then, to exit the program environment, press:
In RPN or algebraic mode: ¹£
Store an initial guess for X of 10 into the variable X. Then set the function to X and solve for the value of X.
Note that the SOLVE function is located on the cursor key area at the top of the HP 33S calculator. If you
are unsure of its location, see the image at the end of this training aid.
In RPN or algebraic mode: 10eXºsXÛX
Figure 3
Since you feel this equation has a negative root as well, store a new guess for X of -10 into the variable X.
There is no need toset the functionto X (since it has already been done). Solve for the value of X.
In RPN or algebraic mode: 10zeXÛX
Figure 4
Answer: Roots found for the equation are –2 and +2.
Example 2: Solve for the roots of Y = X2– 7X + 12
Solution: We're looking for values of X such that X2– 7X + 12 = 0. First, we'll enter a program that computes the
value of the function. If a program already exists in program memory with the name of X, then it will need to
be cleared. This can be done by pressing ¹uÕÏ to have the HP 33S display the list of
programs in the calculator and then press Øto step through the program labels. When the label of the
program to be deleted is shown in the display, pressing ¹¡will delete that program from the
calculator's memory. Pressing Åwill then clear the display and allow you to proceed.
In RPN or algebraic mode: ¹£¹ÓXºd
hX)2Ã7¸hXÙ12Ï
ºÔ
Figure 5
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hp calculators
HP 33S Solving for roots
To show the checksum and length of this program, press the following in RPN or algebraic mode. Note that
the symbol Õ means to press the right arrow direction of the cursor key at the top of the HP 33S. If you
are unsure of its location, see the image at the end of this training aid.
In RPN or algebraic mode: ¹uÕϺÎ
Figure 6
If the checksum of the program just entered does not equal 23EC, then you have not entered it correctly.
To clear the checksum display press:
In RPN or algebraic mode: ¹£
Then, to exit the program environment, press:
In RPN or algebraic mode: ¹£
Store an initial guess for X of 10 into the variable X. Then set the function to X and solve for the value of X.
Note that the SOLVE function is located on the cursor key area at the top of the HP 33S calculator. If you
are unsure of its location, see the image at the end of this training aid.
In RPN or algebraic mode: 10eXºsXÛX
Figure 7
Since you feel this equation might have a root larger than this, store a new guess for X of 100 into the
variable X. There is noneed to set thefunction toX (since it has already been done). Then solvefor the
value of X.
In RPN or algebraic mode: 100eXÛX
Figure 8
The same root is returned. This is a good indication (but certainly not foolproof) that there are no roots
larger than +4 for this equation.
To see if there is a root less than +4 for this equation, store a new guess for X of –10 into the variable X.
Then solve for the value of X.
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hp calculators
HP 33S Solving for roots
In RPN or algebraic mode: 10zeXÛX
Figure 9
Answer: Roots found for the equation are +3 and +4. Note that the HP 33S owners manual provides much more
information about providing initial guesses for the SOLVE feature.
Example 3: Solve for the roots of 4X2– 2X – Y = 12
Solution: First, rearrange the equation so that the variable Y is isolated. This is necessary to use the SOLVE function
as we are doing in this training aid. The rearranged equation is Y = 4X2– 2X – 12.
We're looking for values of X such that 4X2– 2X – 12 = 0. First, we'll enter a program that computes the
value of the function. If a program already exists in program memory with the name of X, then it will need to
be cleared. This can be done by pressing ¹uÕÏ to have the HP 33S display the list of
programs in the calculator and then press Øto step through the program labels. When the label of the
program to be deleted is shown in the display, pressing ¹¡will delete that program from the
calculator's memory. Pressing Åwill then clear the display and allow you to proceed.
In RPN or algebraic mode: ¹£¹ÓXºd
4¸hX)2Ã2¸hXÃ12Ï
ºÔ
Figure 10
To show the checksum and length of this program, press the following in RPN or algebraic mode. Note that
the symbol Õ means to press the right arrow direction of the cursor key at the top of the HP 33S. If you
are unsure of its location, see the image at the end of this training aid.
In RPN or algebraic mode: ¹uÕϺÎ
Figure 11
If the checksum of the program just entered does not equal 8DF1, then you have not entered it correctly.
To clear the checksum display press:
In RPN or algebraic mode: ¹£
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hp calculators
HP 33S Solving for roots
Then, to exit the program environment, press:
In RPN or algebraic mode: ¹£
Store an initial guess for X of 10 into the variable X. Then set the function to X and solve for the value of X.
Note that the SOLVE function is located on the cursor key area at the top of the HP 33S calculator. If you
are unsure of its location, see the image at the end of this training aid.
In RPN or algebraic mode: 10eXºsXÛX
Figure 12
Since you feel this equation might have a root larger than this, store a new guess for X of 100 into the
variable X. There is noneed to set thefunction toX (since it has already been done). Then solvefor the
value of X.
In RPN or algebraic mode: 100eXÛX
Figure 13
The same root is returned. This is a good indication (but certainly not foolproof) that there are no roots
larger than +2 for this equation.
To see if there is a root less than +2 for this equation, store a new guess for X of –10 into the variable X.
Then solve for the value of X.
In RPN or algebraic mode: 10zeXÛX
Figure 14
Answer: Roots found for the equation are –1.5 and +2. Note that the HP 33S owners manual provides much more
information about providing initial guesses for the SOLVE feature.
The next page of the training aid contains a picture indicating the location of the cursor keys and functions used in this
document.
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hp calculators
HP 33S Solving for roots
Image of the cursor key area of the HP 33S calculator
The arrow by the number 1 indicates the location of the cursor key that should be pressed to move to the right, as
indicated by the keystroke Õ.
The arrow by the number 2 indicates the location of the cursor key that should be pressed when the Ûsymbol is
found in the instructions.
1
Figure 15
2
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