Basic methods to solve equations

Transcription

1 Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here: How to solve most basic types of equations. What do think that solving an equation means? Finding what is, of course! But we know already that is a letter of the alphabet, so that cannot be it! OK, I am pulling your legs, but I am doing that to bring up an important point. Most people have an incomplete and out-of-focus idea of what solving an equation is. This incomplete vision may allow them to play the game enough to complete high school, but not to really understand equations and thus having good control on them for more advanced studies of mathematics. So, I will begin with some very basic clarifications. If you are already familiar with them, just move on, but make sure that you are familiar enough that you will not get lost on the net page! Here is some jargon that will be used almost every day: make sure to use it properly! Definition An equality is a mathematical statement claiming that two quantities are equal to each other. An equation is a mathematical statement requiring two epressions containing constants and variable(s) to be equal to each other. However, the statement may be true only for some values of the variable(s). An identity is an equation that is true for any value of the variable(s). An equation can be read as a regular sentence of the form This equals that, with the words equals being represented by the symbol =. Eamples: The statement 4 7 can be read as three plus four equals seven and is an equality. The statement 4 1 can be read as three plus four equals twelve and is a falsehood, since it is not true! The statement 7 can be read as three plus the square of a number equals seven and it is an equation. It is true if the number represented by is or -, but it is false otherwise. The statement 4 7 can be read as three times a number plus four times that same number equals seven times that number. It is an identity, since it is true no matter what number we use for. Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 1

2 The statement 4y 7 is also an equation, true for some combinations of and y, false for others. You should now be able to see what solving an equation actually means. Definition Solving an equation means finding all values of the variable(s) for which the equation is true. Any value of the variable or combination of the variables that makes the equation true is called a solution of the equation. Eample: 4 0 This equation is true if or if, since in that case the left and right side are actually equal. But if we use any other value for, then the statement will not be true. Therefore we have solved the equation and found that and - are its solutions. Some equations can be solved partially, that is, it is possible to find some solutions, but not all. Finally, for some equation the solutions can be only approimated and there are several more or less efficient methods for doing that. When is it useful to solve an equation? Many applied problems are solved by requiring two quantities to be equal. The two quantities may represent what we have and what we want, or two different ways of describing the same thing, or two quantities that must balance each other in order for things to work. In any of these cases the required equality can be represented by an equation and solving that equation means finding conditions for which the desired equality is achieved. Therefore solving an equation is useful when we are trying to find an appropriate way, or sometimes the best way to do something. Remember that even if you are dealing with an abstract equation, that is, one that is not eplicitly associated with a real applied problem, such a problem, or a similar one, is very likely to eist. In fact most of the modern technology on which we depend is related to one or more equations for its correct functioning. So, be encouraged to always ask for eamples of applications of the methods you learn, but trust that such applications do eist, you will see them and it is therefore worth preparing for them. So, how do we solve an equation? As I said, many methods have been devised to solve certain types of equations. In this section I will introduce you to the basic methods; those used most often, and highlight the general mathematical principles behind them. And here are the first two principles, which always go together. I can see the solutions in this case, but how can we find them for any given equation? That is what this section will clarify. We shall review the most basic methods for solving an equation that involves a single variable. Equations involving more than one variable will be seen elsewhere. Is it always possible to solve an equation? Some types of equations are simple enough that routine methods have been devised to solve them eactly and completely. For instance, linear and quadratic equations are of this kind. Other types of equations are such that they can be solved in some cases, but not in general. Polynomial, eponential and trigonometric equations are of this kind. Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page

3 Strategy for solving an equation To solve an equation, try to change it to a different, simpler equation that has the same set of solutions. We may refer to this process as manipulating an equation to find its solutions. The main goal of such manipulations should be to eliminate any nuisances coming from difficult operations or complicated sets of simple operations. If a change may be useful, but leads to an equation with additional solutions, check which of the solutions found are solutions of the original equation. Strategy for manipulating an equation properly: The golden rule of equations When manipulating an equation always do to the left side whatever you do to the right side. Make sure that any step you do is done to the whole sides, not to the individual terms or factors they contain. With this in mind, we are ready for the first, most basic, most common, and most useful method. Strategy for solving equations by isolating the variable To solve an equation, try to manipulate it so that the one side (usually the left) will consist only of and the other side (usually the right) will consist of an epression containing only constants. When that is done, the solution(s) can be read from this simple statement. Eample: 10 4 To solve this equation we start by subtracting 4 from both sides, thus moving the 4 to the left side: The we divide both sides by -: 6 6 Therefore is the only solution. Eample: 10 1 To solve this equation, we start by adding 1 to both sides: 10 1 Then we square both sides to eliminate the nuisance given by the root: 10 ( 1) But since squaring eliminates negatives, this equation is not fully equivalent to the original one, as it also includes the solutions of the equation Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page

4 10 1. Therefore at the end we shall need to check the solutions we find. Now we epand the square on the right: 10 1 Then we subtract 1 from both sides to shorten the formulae: 9 We have now isolated the variable and can obtain all possible solutions: Notice that I introduced a, again to account for the fact that squaring eliminates negatives. But are both values solutions of the original equation, or of the additional one we added by squaring? Let us check: If Since this is true, the solution = is acceptable. If Since this is false, the solution is not acceptable. Eample: 5 We start by noticing that if we end up with a possible solution of 0 we must reject it, since it would make the fraction on the right undefined. Maybe we won t, but now is the time to notice that. To eliminate the nuisance of a denominator, we multiply both sides by 5 : Net we subtract from both sides to reduce the instances of the variable: We can now see that the left side is positive, because of the square, but the right side is negative! This means that this equation has no solutions, a fact that is always a possibility: not every problem in life can be solved. Eample: 5 We subtract from both sides: 8 In order for these two powers of to be equal, their eponents must be equal, so that we can conclude that is the required solution. If you are familiar with logarithms in base, you can use them to eliminate the eponential and isolate the variable, thus finding the same solution. But we shall investigate logarithms soon enough. Eample: 1sin 0 We can apply this method to trigonometric equations as well, provided we known enough special values for them. So, if we subtract 1 from both sides and then divide both sides by, we get: 1 1 sin 0 sin 1 sin The two basic values of whose sine is are,, but we 6 6 must remember that more solutions are found by adding multiples of. Therefore the whole set of solutions can be written as: 7 11 k, k 6 6 I am assuming that you are familiar with trigonometric functions, but if you are not, a later section will allow you to review them Sometimes the variable cannot be isolated by using the simple steps shown above. In that case another basic mathematical principle comes in handy. Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 4

5 Knot on your finger: The zero property of the product A product of several numbers equals 0 if and only if at least one of the numbers is 0: a a a 0 or a 0 or a 0 This property leads to a very general and very efficient strategy, although do not be fooled into thinking that it works every time. In fact, in order to use it, we need to factor the epression that generates the equation, something that cannot always be done eactly. But when it works, it works well, so, here it is. a n n Strategy for solving an equation through factoring If the variable of an equation cannot be isolated, try the following steps: 1. Move all terms to one side (usually the left), so that the other side is 0.. Factor that side as much as possible.. Split the resulting equation into several smaller ones by setting each factor equal to Solve each of the easier equations obtained in this way. 5. Check that all solutions are acceptable. Notice that the heart of this strategy is what I call the salami technique, namely the reduction of a big complicate problem to several simpler ones. This is a very powerful method that you will see again and that I recommend you practice as much as possible, so as to add it to your arsenal of effective methods. Eample: To solve the equation, we move all terms to the left side: Then we factor by grouping: ( ) 5 ( ) 0 Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 5 ( )(1 5 ) 0 Now we split into smaller equations 0; 0; 1 5 0

6 And finally solve each of them: 0; ; 1 5 You can check that all these solutions work: that is because during the solution step we did not do anything that may add unacceptable values as potential solutions. 6 4 Eample: As before, we move all terms to the left Then combine into a single fraction Notice that 0 would not be an acceptable solution because of the denominator. Notice also that a fraction can only be 0 if its numerator is 0, so we focus on that and factor the numerator by grouping: ( ) 0 Then split into smaller equations: 0; Finally, we solve each of them: 0 ; / Eample: e e 0 We collect the common factor: e ( 1 ) 0 Split into smaller equations: e 0; 1 0 Notice that the first equation has no solutions, since the natural eponential is always 0 (a fact we epect you to know, but is reviewed in another section), so the only solution comes from the second equation, namely 1. Eample: sin sin We move all terms to the left: sin sin 0 Then collect the common factor: sin (1 sin ) 0 Then split into smaller equations: sin 0; 1 sin 0 Then solve the easier equations: 0 k ; k Notice that in order to complete the last two eamples we need knowledge about eponential and trigonometric functions respectively. This specific technical knowledge is often required, but does not change the essence of the strategy. Notice also how important the factoring methods are to implement this strategy. While the golden rule and factoring are general strategies that work in many different situations, there is no guarantee of success by using them. Therefore, other special methods can be quite welcome. The most famous and most frequently used such method is the quadratic formula. Can this method be used even when we the equation involves more comple pieces, such as eponential or trigonometric epressions? Certainly: the method works regardless of what the factors are. The complication may arise when solving for the individual factors, though. Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 6

7 Technical fact The quadratic formula For any numbers a, b and c, the solutions of the quadratic equation a b c 0 are given by the formula: b b 4ac a This is a very famous formula and any mathematically educated person is epected to know, understand and use it. Its proof is based on the method of completing the square, which is presented in a later section. Eample: We begin by moving all terms to the left: 0 Now that we have the standard form we use the formula: Eample: 4( ) Same method on the standard form of the equation, namely. 4 1( ) 17 0 Notice that the two solutions of the last eample are irrational and it would be practically impossible to get them by trial and error, or by using some other method you may have seen in school. Speaking of school methods, notice that I will not mention here those methods that you may have seen in high school and that rely too much on guessing and testing. They are inefficient when used in the contet of calculus, so use them if you like them, but do not rely on them eclusively. Eample: Same method: Notice that in this eample we have a single solution, instead of the two that can be generated by the quadratic formula in general. Sometimes this is reported as saying that the equation has a repeated solution, or a solution of multiplicity. Eample: And again: Since the quantity under the square root is negative, this formula is not computable as a real number. Therefore this equation has no solutions. As we have seen already, this can happen and the quadratic formula easily helps us reach the conclusion even in such cases. There is a less well-known consequence of the quadratic formula that you should know. Since it is really just a special case, here it is as an eample. Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 7

8 Eample: This not a quadratic equation, but it becomes one by a change of variable. If we let z, the equation becomes: 5 0 z 5z 0 Because of this rearrangement, this type of equation, which is quadratic in, is called biquadratic. Now we solve the latter equation by using the quadratic formula: z 6 6 1/ Now we need to get back to. Notice that the second value is impossible for us, since z and a square cannot be negative. That leaves us with the solutions. We start by using the basic Pythagorean trigonometric identity: 5 1sin 8sin 1 5 5sin 8sin sin 8sin 1 0 5sin 8sin 4 0 This time we let z sin to get: 5 sin 8 sin 4 0 5z 8z 4 0 Now we apply the quadratic formula: sin / 5 The first value is impossible, since the sine function is never higher than 1. 1 The second value leads to the basic solution sin 0.4 there to all others:, and from 1 1 sin 0.4 k ; sin 0.4 k Incidentally, the method of change of variable, or substitution, is another big tool in all of mathematics. You will see it in action repeatedly. Here it is again: Eample: This time we let e 5 e z e so that: e 5e e 5 e 0 z 5z 0 We apply the quadratic formula and obtain by using natural logarithms: ln1 0 e ln1.5 If you are not familiar with logarithms, trust me and get ecited about learning their properties soon! Eample: 5cos 8sin 1 And what if none of these methods works? If none of the methods described so far works to solve a given equation and you desperately need a value for the solutions of an equation, you still have two options. You may want to search the web for a suitable method. For instance, did you know that there are cubic and quartic formulae, that is, formulae that allow us to solve any third or fourth degree equation? They are quite complicated and involve comple numbers, but they eist and you can find them, for instance on Wikipedia (which is where the links will get you). However, there cannot be any general formula for polynomial equations of degree 5 or higher, although special formulae for special cases do eist. So, what can we do for equations that do not have a general method of solution? We can get an approimate value. Several algebraic methods eist to get a good approimation, but they will not be useful to you for while, so we ll leave them to your future studies. Also, a good graphing calculator can be used, but at this point I want to encourage you to develop good algebraic skills. Remember that a calculator can be misleading: learn to use it with caution. So, instead, here is a good variety of activities on which you can practice the equation-solving methods I have presented. Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 8

9 Summary The general strategy for solving equations is based on two principles: isolating the variable and factoring in order to split the equation into smaller, simpler ones. When factoring an equation in order to use the zero property of the product, first move all terms to the left side, so that the right side is 0. Several method to solve special types of equations eist, the most famous one being the quadratic formula that solves every quadratic equation. Common errors to avoid Do not make arbitrary changes to an equation, but only those that fulfill the golden rule and obey some proper mathematical rule. Avoid falling prey to the desire to use tempting short cuts. They may be short, but are likely wrong! Learning questions for Section P 1- Review questions: 1. Briefly eplain what solving an equation means.. Describe how to use the zero property of the product to solve an equation.. Describe how to use the method of isolating the variable to solve an equation. 4. Eplain when and how we can solve an equation by using the quadratic formula. Memory questions: 1. When is a product of several numbers equal to 0?. Which basic algebra process is fundamental when solving equations?. Present the quadratic formula for the equation a b c 0 Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 9

10 Computation questions: For each of the equations presented in questions 1-66, perform all steps needed to find its solution and, if possible, identify all such solutions. If the equation requires the use of some transcendental function with which you are not yet familiar, wait until you study it to complete the solution, or ask your instructor for how to handle it e e / / ( 1) 5( 1) ( ) ( ) ( ) 4 ( ) / / 5/ 5 0 4/ 1/ / / Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 10

11 / 1/ 4 / 4 / 1/ / 4 0 1/ 5 4 / 5 8 / 5 / 5 / e e e 4e 5 6e 5e sin cos 0 sin sin 0 5. sin sin 54. cos 4cos sin cos sin cos cos 1 cos sin 1sin e e e 4 e e e Theory questions: 1. Which basic formula, useful in the solution of the main question, is based on the method of completing the square?. The left side of the given equation consists of all powers of. Why does this NOT imply that =0 is a possible solution?. How are factoring and solving equations related? 4. Apply the quadratic formula to the equation a b c Is it possible for an equation to have infinitely many solutions? Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 11

12 Proof questions: 1. Find the eact value of the number described by the epression..., with the sequence of radicals being infinite.. Given a positive number a, find the eact value of the number described by the 1 1. If, show that the value of can be computed without solving the equation for, and determine such value. Can you also find the solution(s) of the equation? epression a a a a..., with the sequence of radicals being infinite. Templated questions: 1. Construct a (relatively simple) equation and try to solve it. If none of the methods discussed here work, use an approimate method, such as a calculator, and start eploring further methods that may be useful for it. What questions do you have for your instructor? Prerequisites for Calculus Chapter 1: Algebra Section : Solving equations Page 1