1. A definition of Algebra: a branch of mathematics which describes basic arithmetic relations using variables.

Transcription

1 MA30S PRECALC UNIT C: ALGEBRA CLASS NOTES. A definition of Algebra: a branch of mathematics which describes basic arithmetic relations using variables.. It is just a language. Instead of saying Jason is 6 years older than Karen ; a mathematician says: J = K + 6. It is a language that is universal and very concise with a select few but well defined rules. For eample we could change the equation to K=J 6 because of one of the special rules. 3. To date we have mainly just used Algebra of simple functions like lines. In this unit we learn to manipulate more comple relationships between variables. QUADRATIC EQUATION 4. The general form of a quadratic function is of the form y = f() = a + b + c. When the function takes on a specific value for y, it becomes an equation. The most common equation of interest is when the y = 0. To solve means to find 5. The quadratic equation is of the form a + b + c = 0 a value of a variable (or variables), that makes an equation true. Try to solve the equation 5 6 = 0. Put your answer below. The solution is the root(s) of the equation or the -intercepts of the function graph. We had avoided finding the roots or -intercepts in Unit A. Recall also that a quadratic function can have either 0,, or roots or -intercepts. SOLVING QUADRATIC EQUATIONS USING GRAPHING TOOLS 6. We already did this in Unit A. Just graph the function, find where it crosses the -ais. Using the CALCULATE function of the TI-83 is the easiest method. Notice this often only gives an approimate but very close answer though! Really close, but not necessarily eact! MA30SP_C_AlgebraNotes.doc Revised:03005

2 SOLVING QUADRATIC EQUATIONS USING FACTORING 7. Back to Grade 9! Eample: solve for given that = 0. Recall how to factor: what two numbers multiply to give 9 but add to give 6. The result is that can be factored as ( - )( + ). So if = 0 is the same as ( )( + ) = 0then it can only be true for = and = - 8. The factoring method works nicely for some nicely contrived quadratic epressions and for integer number solutions. You will need to recall your factoring of trinomials and difference of squares from Grade 9 and 0. a. Review of Factoring of polynomial epressions: a + b + c. Factoring actually seldom works in real life applications unless the question has been nicely contrived. You will soon learn away that you never have to factor again! b. Factoring General Trinomials a + b + c, where a =. c. Factoring Perfect Squares. Special trinomials where (b/) = c and a = Perfect squares are numbers like:, 4, 9, 6, 5, 36, 49, etc. Eample: What two factors multiply to give 6 and add to give 7. 6 and. So is the same as ( + 6)( + ) Has two different roots Eample: The first term is a perfect square and so is the last. Good clue that you may be able to just write: ( + 3 )( + 3 ) or just ( + 3 ) It has two factors the same, the equation equated to 0 has only one root: 3.

3 3 d. Factoring Difference of squares. They have no b term (it is zero). A typical form: a c where a is a perfect square and c is a perfect square. e. Factoring when a These are often considerably more difficult and only work for specially contrived epressions. Methods: Try factoring out the a first Try Trial and error The AC Method. See the appendi to these notes for the AC Method. Eample: 4 8. Since this is the difference of two perfect squares it can immediately be written as: ( + 9)( 9) (Has two separate factors and therefore two separate roots of its equation: 4.5 and +4.5) Eample: 6 + Factoring out the a. 6 + = 6 ( + /6 /6) What two things multiply to give /6 and add to give +/6? Can you see that it is ½ and /3? I don t epect that you do! But sometimes factoring out the a does work nicely if it can be done. Trial and error. You know it should look something like (m + p)(n + q), so that m*n=6 and p*q = and mq + np = + Notice ( + ) (3 ) works nicely 9. You try a few; factor each of the following quadratic equations and state the roots (without using a graph or graphing calculator). Recall that a quadratic has a square on one unknown. Notice also that we always equate a quadratic equation to zero to solve it! a. 5 6 = 0 b = 0 c. 4 4 = 0 d. 4 6 = 0 e. = 4 f = 0

4 4 e. = 4 f = 0 0. Now check the solutions (the roots) with a graphing calculator or an on-line grapher to see that they are correct. AN APPLICATION. An object is shot upwards at a speed of 300 meters per second on earth. The equation for its height above ground is h(t) = 5t + 300t. The h, a function of t, is height in meters, the t is time in seconds. Find when then the body hits the earth after it is thrown; ie: when h = 0. Ans: Solve for 0 = 5t + 300t. So t = 0 or t = 60. Solving Simple Radical Equations by Unsquaring (taking Square Root). A radical equation involves roots (square roots, cube roots, ) and fractional eponents. We will only need square roots in Grade. Solve for : = 4; = ; = 9; =. = 8; = 3. Notice that we always have answers when we take a square root (or any even root for that matter). We use the symbol toshow the answers, it really just means Solve Simple Radical Equations by Taking square Root: Eample: ( + ) = 64 Unsquare Both sides: + = ± 8 Remember there are two answers when unsquare! Isolate the : = ± 8; so = 0 or = 6 Solution set: { 6, 0}

5 5 You try; solve for : a. ( ) = 6 b. ( + 3) + = 8 Ans: {-, 6} c. ( 4) = 70 Ans: {-7, } d. ( ) = 4 Ans: {-, 0} Ans: { + 7; 7} or ± 7 SOLVING TRIGONOMETRIC EQUATIONS WITH A QUADRATIC FORM Eample : Solve for θ [0,360]. (if you have not done Grade Trig yet omit this) cos (θ) = 0 Note: cos (θ) means (cos(θ)) Ans: 45, 35, 5, 35 Eample : Solve for θ [0, 360]. 9sin θ = +4 Notice there are four answers to these quadratic trigonometric equations! Ans: θ = 4.8, 38.,.8, 38.

6 6 4. If you find trigonometric functions in a quadratic form confusing just make them easier to understand by substituting something simpler for the trigonometric function. So cos (θ) = 0 becomes = 0. Then don t forget to turn the back into cos(θ) at the end. IMAGINARY NUMBERS 5. We have been told it is not possible to take the square root of a negative number. That is not quite true! Several hundred years ago we invented a number system called the imaginary numbers. 6. The imaginary number i is defined as: i = or i = + It is used to write the square root of any negative number 7. i has the property sufficient to solve: a = * a = * a = i a it also has the property: Eample: 8. Solve = 8. Solution: = ± 8 = ± 4 = ± i ( i a ) = a Don t panic over the ±. It is just a mathematicians way of saying two things at once. It is not really an actual operation! It means there is a plus answer and a negative answer SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE 9. Not all quadratic equations are easily factored by just looking at them. And not all are in a simple squared binomial form; ( h) ready to unsquare. Often Completing the Square will help. Recall completing the square from Unit A Quadratics. We add and subtract a magic number to make a perfect square trinomial. Completing the square is the way we can take equations with an and an in them and just turn them into something with an just an + h in them that makes them easier to solve.

7 7 EXAMPLE : (simple where a in the general form is equal to ) 0. Solve for : 4 + = 0 Solution: Remember? The majic number to add is half of the b coefficient of the, all squared. Or (b/) if and only if the a is. 4 + = 0 4 = = = + 4 ( - ) = = ± = ± Check the positive answer (+ ) 4(+ )+= = 0 General form Isolate the variable terms to one side b Add to both sides. (This is the same a adding and subtracting it on the same side) Simplify Factor and simplify Check that the answers work by evaluating! Check the negative answer:. Check the answer with a graphing calculator or an on-line graphing tool by graphing now if you want! But of course, it will only give you a real close answer, not eact like when we solve it analytically using algebra.. Check the answer also just by evaluating a very accurate value of your solutions. Knowing how to use the memory button on your calculator will help to remember those loengthy approiamtions of the irrational values.

8 8 EXAMPLE : (where the a in the general form is not equal to ; a ). Solve: 6 8 = 0 Solution: The secret to these, where the leading coefficient on the term is not a (a ), is just to divide the entire equation by that annoying a coefficient if you can. 3. Check the answer with a graphing calculator or other graphing tool. THE QUADRATIC FORMULA 4. All the previous methods of solving a quadratic are interesting but they have their limitations. There is a way to solve a quadratic everytime with one formula. It is called the quadratic formula. It looks like this: b ± = Memorize it if you plan to be in science or engineering! b 4ac a It is really just a pre-washed version of completing the square! DERIVATION AND PROOF OF THE QUADRATIC FORMULA 5. The quadratic formula is easily derived from the general form or a trinomial. The quadratic formula is really just completing the square, with all the algebra already done for you so all you have to do is plug in. (You are not responsible to know the proof, but interesting to see why it is true! It simply falls out of completing the square!) (See separate notes). If you take Calculus in university you will do lots of proofs!

9 9 EXAMPLES OF SOLVING WITH QUADRATIC FORMULA 6. Solve for : = 0 How many roots does this have? 7. Solve for : =0 How many roots does this have? SOLVE EQUATIONS THAT CAN BE RE-WRITTEN AS QUADRATIC EQUATIONS 8. Some equations are not really quadratic, but some can be re-written as such. Solve: 4 5 = 0 This is called a quartic equation, its highest power is 4. To solve, just say something else =

10 0 9. Some trigonometric equations have a quadratic form. Solve for θ: sin θ + sinθ = 0 by substituting = sin(θ) and using the quadratic formula. Keep θ in the interval domain of [0, 360] Ans: 30, 50, 70 (Omit this idea if you have not yet done Grade Trig. But you will see it again) PROPERTIES OF THE QUADRATIC FORMULA 30. Discriminant. The epression b 4ac is called the discriminant. It separates or discriminates how many roots and what type of roots a quadratic formula has. If b - 4ac is: Then the equation has: >0 and is a non-perfect square real irrational roots >0 and is a perfect square real rational roots =0 real number root <0 0 real roots; the roots are imaginary

11 3. Graph the following equations, and then check with the discriminant, (b 4ac), to see how many real number solutions eist. Record the number and type of roots for each eqn: a) = 0 Nbr and type of Roots: b) = 0 Nbr and type of Roots c) 4 = 0 Nbr and type of Roots d) + = 0 Nbr and type of Roots e) = 0 Nbr and type of Roots f) + - 4=0 Nbr and type of Roots g) = 0 Nbr and type of Roots h) = 0 Nbr and type of Roots i) = 0 Nbr and type of Roots Often in engineering it is handy to check the discriminant first to make sure there will be a solution before you start calculating the full solution. WRITE THE QUADRATIC EQUATION GIVEN ITS ROOTS 3. If a quadratic equation has two roots: r and r, then the quadratic equation can be written as the product of two binomials: ( - r )*( - r ) = 0 EXAMPLE : Say the roots are known to be 3 and +4. Then the quadratic can be found from the two binomials: ( + 3 )*( 4) = 0. or by multiplying: = 0

12 EXAMPLE. The roots (ie: where y = 0) are known to be and The quadratic equation can be re-written as: * = * = by F.O.I.L = 0 Simplifying = 0 Further simplifying = 0 is one quadratic equation with the roots and Notice also though that multiplying by 3; = 0 is also a quadratic equation that has those roots. There are an infinite number of equations that have the same roots. You would need another point (or a verte) to get a particular equation. EXAMPLE OF FINDING ROOTS 35. It is known that the equation for the morning rush hour traffic at Higgins and Main can be closely modeled by: c = 45t + 750t 900 where c is the number of cars in a minute and t is the local time in hours. The equation is valid only throughout the domain 06:00 to :00. The construction crews don t like to disrupt traffic when traffic eceeds more than 00 cars in a minute. Find the approimate times between which the road crews should avoid disrupting traffic. Solution: 36. We want to know when 00 = 45t + 750t 900. So find the roots of the equation: 0 = 45t + 750t 3000

13 3 EXAMPLE FROM THE DAVINCI CODE 37. Maybe you watched the movie or read the book the Davinci Code. A key part of that movie is the Divine Number or the Golden Ratio. The number they give in the movie is.68, but they are wrong: it is actually the positive solution to the equation = You find the magic Divine Number from the movie eactly!!! None of this decimal or rounding stuff! Only eact numbers! Even sun flowers grow obeying this equation! SOLVING ABSOLUTE VALUE EQUATIONS ABSOLUTE VALUE DEFINITION 39. The absolute value of a number can be thought of as its distance from 0 on a number line. Any positive number and any negative number are each a positive distance from 0. The absolute value is concerned with distance, not direction The absolute value of any real number a, written as a, is defined in two parts as: and a a = a if a 0 = a if a 0 In other words, just think of the as being like magic brackets that strip away any negative sign inside them. 4. Eample: There are two values that have a distance of from zero. They are and. So if =, then = or = -. Distance is Distance is 0 -

14 4 4. Practice. Evaluate the following epressions: a. 4 = b. 4 = c. 3 5 = d = e. Given = : 4 = f. Given = 5: 4 = g. 4 = h. 3 4 = GRAPHING THE ABSOLUTE VALUE FUNCTION 43. The absolute value function,, tends to preclude negative values and puts in abrupt changes in direction in a graph. This is kinky! 44. Try manually graphing these three functions on the same graph. a. f() = 3 3 b. g() = 9 Do not scale the graph here, if the value of the function is too large, just do not plot it f() g() Find out how your calculator and also the TI-83 graphing calculator does the absolute value function. It usually looks like ABS( ) on a calculator or in a spreadsheet function. Interesting to note: the ABS() or function can always be duplicated by doing the positive root of the square of a number. So is identically the same as. Try it!

15 5 Graph the last two equations above with a graphing calculator or an on-line graphing tool to see that you get the same graphs. SOLVING ABSOLUTE VALUE EQUATIONS 39. EXAMPLE : Solve and Graph 3 =. So, by definition of the absolute value function, () when 3 is 0 we have 3 = ; but () when ( 3) is <0 we have ( 3) = ; or 3 = We really have two equations: 3 = and ( 3) = So = 5 or =. But! ******* It is always important to check the answers in the original equation to find any etraneous answers. Etraneous answers are answers that seem to be an answer, but when you check they really are not. Does 5 3 =? Yes! Does : 3 =? Yes!. So both answers are proper solutions

16 6 EXAMPLE Absolute Value Equation Solution 40. Solve: 3 = Again. We actually have two equations 3 = and 3 = ( 4 + 6) = Solve by factoring or completing the square or using the quadratic formula..we get two intermediate answers from each equation. But don t forget to check them! Two of them are etraneous and are therefore wrong. 4. Now solve it by graphing it! You will see that there are only two intersections (or solutions ) where the graphs have the same points. 43. Every problem in the universe can be solved by graphing! Of course, if there are more than 3 unknowns you need to plot in more than 3 dimensions! Therein lies the problem! Absolute Value Equation Eample ^-3X The absolute value is a kinky function!

17 7 SOLVE RADICAL EQUATIONS 45. A radical equation is one that contains radicals or rational eponents. To solve the equation, try to eliminate the radicals and obtain a linear or quadratic equation. 46. The following property helps: n If a = b then a = b n 47. In other words both sides of an equation can be raised to the same power. [caution though, range and domain can change!!] EXAMPLE 48. = 4 ) 4 = 6 ( =. And check answer! EXAMPLE / = 8 3 / / 3 / 3 3 ( ) = 8 = 8.. etc.

18 8 EXAMPLE = gives = 7 and =. But is etraneous! Always check your solution. Try graphing this on the TI-83 or a graphing tool on-line and see that an -value of does not eist as a solution! (for real numbers at least). But = 7 works as a solution sqrt(4+8) Graph of = SOLVE RATIONAL EQUATIONS 5. An equation involving rational epressions is called a rational equation. A rational epression has a variable (or variables) in a denominator. A rational epression is not a like a nice polynomial (like y = a ) at all. The best way to solve is to multiply both sides by the Least Common Denominator (LCD). A Least Common Denominator is just a Least Common Multiple (LCM) but in the denominator. You may recall LCMs from earlier studies. If you do not recall LCMs from earlier studies check out the appendi to these notes. 5. The LCM of two or more polynomials is the simplest polynomial that is a multiple of each of the original polynomials. 53. To find the LCM Factor each polynomial Use each factor the greatest number of times it occurs in each polynomial. EG: LCM of 4 and 6. * and *3. So the LCM is **3=. EG: LCM of 9 and 3 ( 3)*( + 3) and ( 3). So the LCM is ( 3)*( + 3)

19 9 49. The Least Common Denominator (LCD) of two or more fractions is the Least Common Multiple of the denominators of the fractions. 50. To solve a rational equation, you must first identify which answers are forbidden due to undefined answers dividing by zero. Then multiply each term on both sides by the LCD. Make sure the denominators are factored before you commence solving. EXAMPLE : Solve: 3 6 = + Note cannot =! LCD =. Multiplying though by the LCD.gives =. Check. Note that is not allowed when checking since can t divide by 0! No solution 30 EXAMPLE : = Note first that 3 is definitely not a legal answer! Neither is 3 LCD is 9 so multiply thru by that.. Answer is = with = 3 being an etraneous root. 3 + EXAMPLE 3. = + Note cannot equal LCD = ( )( + ). So multiply through. Answer is = and = 5; no etraneous root.

20

21 A- APPENDIX FACTORING USING AC METHOD. Factoring trinomial epressions in one unknown is an important skill necessary to eventually solve quadratic equations. Trinomial epressions are of the form a + b + c and they can sometimes be factored into two binomials using real numbers. You will learn later that there are other numbers that are not real, and that in fact all trinomials can be factored into two factors. And don t forget: factoring is just un-multiplying! In fact, factoring is really just dividing in other words.. You likely have learned several methods already: factoring out common factors, perfect squares factoring, difference of squares factoring, grouping, trial and error, etc. 3. Factoring epressions of the form a + b + c can often be complicated if the a (a not equal to one). 4. Eample: Try your known methods of factoring the trinomial epression: a. You can t do the simple trick of finding two integers that multiply to give 6 and add to give 5 because the a (a = 4) is not equal to one. b. You might stumble upon the grouping method: and notice that could be written as: 4 ( 3 +8) 6 so: so: (4 3) + (4 3) which is bouquets of (4 3) plus bouquets of (4 3) so: ( + ) bouquets of (4 3); so ( + )(4 3). Notice I like to call bracketed terms bouquets because that is what they really are! My wife loves her bouquets of 4 s less three. Last week I bought her of them and this week I bought her of them. So she got ( + ) bouquets of (4 3) or

22 A- THE AC METHOD OF FACTORING 5. The AC Method will work some times for many trinomial polynomial epressions to help you more easily find the four terms that can be readily factored by grouping. Let s try Step : Multiply the a*c: ac = 4 Step : Find the factors (that multiply together) of 4 and that add to give +5. Notice that 3 and +8 work. Step 3: Rewrite the original epression with the smaller number times first (without consideration of its sign; the one with the smaller absolute value as we call it) and the larger number times net in place of the original term; Notice that is still 5. Step 4: Factor the first two terms: (4 3) and factor the second two terms (4 3) Step 5: Do the final factoring. (4 3) + (4 3) both have (4 3) in common; so (4 3)( + ) is the final answer. Step 6: Check your answer by multiplying out again (F.O.I.L) (4 3)( + ) = PRACTICE EXAMPLES: 6. Factor: ac = 7 which has factors of 36 and, 9 and 8, etc. 9 and 8 add to give 7 though! So rewrite as: (smallest factor: 8 first) Factor the first two terms and the last two terms: (3 + 4) + 3(3 + 4) Do the final factoring since both parts have a in common; thus: (3 + 4)*( + 3) Check the answer by multiplying!

23 A-3 7. Eample : Factor 0 3 ac = 30 which has factors of 5, and 5, 6 and 6, 5, etc... 6 and 5 add to give the b of though! So rewrite as: Factor the first two terms and the last two terms: 5( + ) + -3( + ) Do the final factoring since both parts have a in common; thus: (5 3)*( + ) Check the answer by multiplying! It works! 8. One for you to try. Factor: (hint: 3 and are the ac factors) The answer is: (8 + )( + 4) but notice you should have factored out a GCF constant factor of two first to make the factoring less onerous, so really the fully factored answer is: (8 + )( + ). Multiply it out to check it! FACTORING IN TWO UNKNOWNS 9. Factoring in two unknowns is also possible using the AC Method with some adaptation. 0. Eample. Factor: + 5y + y. Pretend this is in a form a + by + cy ac = 4 and factors 4 and add to 5 so: + y + 4y + y. So: ( + y) + y( + y) So: ( + y)( + y)

24 A-4 SOME PRACTICE ON YOUR OWN 7. Try these on your own. The answers are given. Factor: a b. 5 7 Ans: (3 + 4)( + ) c Ans: (3 )(5 + ) d Ans: (6 + 7)( + 3) Ans: (5 + )(3 0) Simplify by factoring the numerator e. + 6 f Ans: ( + 3) Ans: (3 + )

25 B- APPENDIX B FACTORING USING GEOMETRY (ALGEBRA TILES). Some students like to see a more visual way of factoring. Your nieces and nephews in Grade 5 and 6 probably learn it this way using Algebra Tiles. In fact this is the original way that ancient mathematicians factored trinomials. This is just a demonstration of the method, you can research it further on your own if you want.. Factor the trinomial The ancients would draw squares of size * (ie: ) in the dirt, and rectangles of length and width one, and tiny squares of length one by one. You have to pretend you do not know how long a length of is; it is rubber. long long wide 4. So would be scratched in the dirt as: 5. And then they noticed they could group them together like this: Which is really two groups of

26 B- 6. But both of these groups could be re-grouped like this: Which can be epressed mathematically as bunches of (+3) by (+); in other words: ((+3)(+)) or (+3)(+) So: = (+3)(+) Of course, negative numbers were a bit perpleing for the ancients! So they didn t bother with negative numbers. But you may find your nieces and nephews are taught that red tiles are negatives, and green are positive and that a red cancels a green because they are opposites. It just complicates things, but it can be done.

27 C- APPENDIX C PRACTICE PROBLEMS Eact Solutions only! None of those crazy rounded off decimals earthling things! Change up the methods if you want! Solve by Factoring a. p + 5p 4 = 0 Don t forget to equate to zero! b. w w 30 = 0 {-7, } {-5, 6} c. u + 9u = 60 d. 4t 9t + 6 = 5 4t u=-4 or u = -5 e. c + 8c + = 0 {, } 8 f. 4y 47 = c = ; c = 6 Solve by Taking Square Root a. 9n + 0 = 9 y = ± 7 b. 3 4 = 85 {± 3} ±

28 C- c. 8t 8 = 0 d. 0 + z = 6 9 t = ±3 e. 3n + 5 = 4 {} f. ( z ) = {± 3} z =± or better: z = ± 3 Solve by completing the square a. w 8w = 65 Always works! b. h + 4h + 3 = 5 {-3, 5} 7 ± 7 c = 3 d = 5 ± or better yet 3 ± 3 3

29 C-3 Solve using the Quadratic Formula a. w 8w = 65 b = 0 c. + = 3 d = 5 Solve the Absolute Value Equations a. 3 = 7 b = 6 c. + 4 = 8 d = 0

30 C-4 Solve the Radical Equations: a. 7 = 4 b. = + 4 = 3 c. y + 9 = y + 3 No solution d = 4 y =5 Solve the Rational Equations = 8 or Check for forbidden answers! a = b. 4 = = -6 = c. = + d. n + 4n = 4n = -0 {-}