Chapter 13 Overview Some More Math Before You Go The Quadratic Formula The iscriminant Multiplication of Binomials F.O.I.L. Factoring Zero factor propert Graphing Parabolas The Ais of Smmetr, Verte and Intercepts Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski The Saga of Mathematics Overview The Quadratic Formula Simultaneous Equations Gabriel Cramer (1704-175) Cramer s rule eterminants Algebraic Fractions Equation of a Circle Remember the quadratic formula for solving equations of the form a + b + c 0 in which a, b and c represent constants. Lewinter & Widulski The Saga of Mathematics 3 Lewinter & Widulski The Saga of Mathematics 4 The Quadratic Formula The Quadratic Formula It is eas to determine these constants from a given quadratic. In the equation + 3 5 0, we have a, b 3, and c 5. At times, b or c can be zero, as in the equations 9 0 and 8 0, respectivel. b ± b 4ac a Usuall ields two different solutions in light of the plus or minus sign (±). The quantit b 4ac is called the discriminant. It determines the number and tpe of solutions. Lewinter & Widulski The Saga of Mathematics 5 Lewinter & Widulski The Saga of Mathematics 6 Lewinter & Widulski 1
The iscriminant Imaginar Numbers If b 4ac > 0, then there are two real roots. If b 4ac 0, then there is one real (repeated) root, given b b/a. If b 4ac < 0, that is, if it is a negative number, it follows that no real number satisfies the quadratic equation. This is because the square root of a negative number is not real! Mathematicians invented the so-called imaginar number i to deal with the square roots of negative numbers. i 1 If one accepts this bizarre invention, ever negative number has a square root, e.g., ( 1) 9 1 3i 9 9 Lewinter & Widulski The Saga of Mathematics 7 Lewinter & Widulski The Saga of Mathematics 8 (continued) Let s solve the quadratic equation 5 + 6 0 using the formula. Note that a 1, b 5, and c 6. The discriminant is ( 5) 4( 1)( 6) 5 4 1 It s positive so we will have two solutions. The are denoted using subscripts: 5 + 1 5 1 1 3 and Lewinter & Widulski The Saga of Mathematics 9 Lewinter & Widulski The Saga of Mathematics 10 Problem Solving (Falling Objects) Multiplication of Binomials In phsics, the height H of an object dropped off a building is modeled b the quadratic equation H 16T + H 0 where T is time (in seconds) elapsed since the object was dropped and H 0 is the height (in feet) of the building. If the building is 100 feet tall, determine at what time the object hits the ground? : Multipl + 1 b 3 The work is arranged in columns just as ou would do with ordinar numbers. + 1 3 6 3 + 5 3 Lewinter & Widulski The Saga of Mathematics 11 Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski
F.O.I.L. F.O.I.L. The first term in the answer,, is the product of the first terms of the factors, i.e., and,. The last term in the answer, 3, is the product of the last terms of the factors, i.e., +1 and 3. The middle term, 5, is the result of adding the outer and inner products of the factors when the are written net to each other as ( + 1)( 3). The outer product is 3, or 6, while the inner product is (+1), or +. As the middle column of our chart shows, 6 and + add up to 5. You ma remember this b its acronm F.O.I.L. which stands for first, outer, inner and last. Lewinter & Widulski The Saga of Mathematics 13 Lewinter & Widulski The Saga of Mathematics 14 ( + 1)( 3) First: F.O.I.L. 6 5 3 + 3 Outer: ( 3) 6 Inner: (+1) + Last: (+1) ( 3) 3 Factoring Let s look at the quadratic equation 5 + 6 0 Can this be factored into simple epressions of first degree, i.e., epressions involving to the first power? is just times. On the other hand, 6 6 1 6 1 3 3. Lewinter & Widulski The Saga of Mathematics 15 Lewinter & Widulski The Saga of Mathematics 16 Factoring Factoring How do we decide between the possible answers ( + 6)( + 1) ( 6)( 1) ( + 3)( + ) ( 3)( ) The middle term, 5, is the result of adding the inner and outer products of the last pair of factors! So the quadratic equation 5 + 6 0 Can be written as ( 3)( ) 0 How can the product of two numbers be zero? Answer: One (or both) of them must be zero! Lewinter & Widulski The Saga of Mathematics 17 Lewinter & Widulski The Saga of Mathematics 18 Lewinter & Widulski 3
Zero Factor Propert Graphing Parabolas Zero Factor Propert: If ab 0, then a 0 or b 0. To solve the equation for, we equate each factor to 0. If 3 0, must be 3, while if 0, must be. This agrees perfectl with the solutions obtained earlier using the quadratic formula. Quadratic epressions often occur in equations that describe a relationship between two variables, such as distance and time in phsics, or price and profit in economics. The graph of the relationship a + b + c is a parabola. Lewinter & Widulski The Saga of Mathematics 19 Lewinter & Widulski The Saga of Mathematics 0 Graphing Parabolas Graphing Parabolas Follow these eas steps and ou will never fear graphing parabolas ever again. 1. etermine whether the parabola opens up or down.. etermine the ais of smmetr. 3. Find the verte. 4. Plot a few etra well-chosen points. If a is positive, the parabola opens upward. If a is negative, the parabola is upside down (like the trajector of a football). a > 0 a < 0 Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski The Saga of Mathematics The Ais of Smmetr The Verte Ever parabola has an ais of smmetr a vertical line acting like a mirror through which one half of the parabola seems to be the reflection of the other half. The equation of the ais of smmetr is b a The verte is a ver special point that las on the parabola. It is the lowest point on the parabola, if the parabola opens up (a > 0), or It is the highest point on the parabola, if the parabola opens down (a < 0). It las on the ais of smmetr, making its - coordinate equal to b/a. Lewinter & Widulski The Saga of Mathematics 3 Lewinter & Widulski The Saga of Mathematics 4 Lewinter & Widulski 4
The Verte The Intercepts To get the -coordinate, insert this value into the original quadratic epression. Finall, pick a few well-chosen values and find their corresponding values with the help of the equation and plot them. Then connect the dots and ou shall have a decent sketch indeed. Sometimes the easiest points to plot are the intercepts, i.e., the points where the parabola intersects the and aes. To find the -intercepts, insert 0 into the original quadratic epression and solve for using the quadratic formula or b factoring. The -intercept is given b the point (0, c). Let s do an eample. Lewinter & Widulski The Saga of Mathematics 5 Lewinter & Widulski The Saga of Mathematics 6 (continued) Graph the parabola 4 5. 1. Since a 1 > 0, the parabola opens up.. The equation of the ais of smmetr is ( 4)/( 1). 3. Plugging this value into the equation, tells us that the verte is (, 9). 4. Solving 4 5 0 will give us the - intercepts of ( 1, 0) and (5, 0). ( 1,0) (0, 5) ais (, 9) (5,0) Lewinter & Widulski The Saga of Mathematics 7 Lewinter & Widulski The Saga of Mathematics 8 Linear Equations a + b c Simultaneous Equations Solve the equation + 10. There are infinitel man pairs of values which satisf this equation. Including 3, 7, and 5, 15, and 100, 110. Then there are decimal solutions like.7, 7.3, and so on and so forth. The solution set is represented b its graph which is a straight line. Simultaneous equations or a sstem of equations is a collection of equations for which simultaneous solutions are sought. : + 10 6 Solving a sstem of equations involves finding solutions that satisf all of the equations. The sstem above has 8 and or (8, ) as a solution. Lewinter & Widulski The Saga of Mathematics 9 Lewinter & Widulski The Saga of Mathematics 30 Lewinter & Widulski 5
Simultaneous Equations Geometricall Recall, the solution set for a linear equation in two variables is a straight line. Geometricall, we are looking for the intersection of the two straight lines. There are three possibilities: There is a unique solution (i.e. the lines intersect in eactl one point) There is no solution (i.e., the lines do not intersect at all) There are infinitel man solutions (i.e., the lines are identical) unique solution no solution infinitel man solutions Lewinter & Widulski The Saga of Mathematics 31 Lewinter & Widulski The Saga of Mathematics 3 Simultaneous Equations Simultaneous Equations When there is no solution, we sa the sstem is inconsistent. There are several methods for solving sstems of equations including: The Graphing Method The Substitution Method The Method of Addition (or Elimination) Using eterminants To solve a sstem b the graphing method, graph both equations and determine where the graphs intersect. To solve a sstem b the substitution method: 1. Select an equation and solve for one variable in terms of the other.. Substitute the epression resulting from Step 1 into the other equation to produce an equation in one variable. 3. Solve the equation produced in Step. 4. Substitute the value for the variable obtained in Step 3 into the epression obtained in Step 1. 5. Check our solution! Lewinter & Widulski The Saga of Mathematics 33 Lewinter & Widulski The Saga of Mathematics 34 Simultaneous Equations To solve a sstem b the addition (or elimination) method: 1. Multipl either or both equations b nonzero constants to obtain opposite coefficients for one of the variables in the sstem.. Add the equations to produce an equation in one variable and solve this equation. 3. Substitute the value of the variable found in Step into either of the original equations to obtain another equation in one variable. Solve this equation. 4. Check our answer! + 10 Solve: 5 14 Since the coefficients of are opposites, we can add the two equations to obtain 6 4. Solving for gives 4. We then substitute this into one of the original equations and solve for which results in 6. So the solution is (4, 6). Lewinter & Widulski The Saga of Mathematics 35 Lewinter & Widulski The Saga of Mathematics 36 Lewinter & Widulski 6
Problem Solving Problem Solving A plane with a tailwind flew 5760 miles in 8 hours. On the return trip, against the wind, the plane flew the same distance in 1 hours. What is the speed of the plane in calm air and the speed of the tailwind? Let speed of the plane in calm air speed of the tailwind Use the formula: distance rate time With the wind rate + time 8 distance 8( + ) Against the wind 1 1( ) This ields the following sstem: 8 + 8 5760 1 1 5760 Use the method of elimination to solve. Lewinter & Widulski The Saga of Mathematics 37 Lewinter & Widulski The Saga of Mathematics 38 Gabriel Cramer (1704-175) eterminants Cramer published articles on a wide range of topics: geometr, philosoph, the aurora borealis, the law, and of course, the histor of mathematics. He also worked with man famous mathematicians, including Euler, and was held in such high regard that man of them insisted he alone edit their work. Lewinter & Widulski The Saga of Mathematics 39 eterminants are mathematical objects which are ver useful in the analsis and solution of sstems of linear equations. A determinant is a function that operates on a square arra of numbers. The determinant is defined as A C B A B C Lewinter & Widulski The Saga of Mathematics 40 Cramer s Rule 4 3 4 3 8 8 + 6 14 ( 6) ( ) Cramer s rule sas that the solution of a sstem such as A + B C + E F can be found b calculating the three determinants, and. These three are defined b the following: Lewinter & Widulski The Saga of Mathematics 41 Lewinter & Widulski The Saga of Mathematics 4 Lewinter & Widulski 7
Cramer s Rule Cramer s Rule C F and B, E A A B E C F, The determinant is called the determinant of coefficients, since it contains the coefficients of the variables in the sstem. The -determinant is simpl with the column containing the coefficients of replaced b the constants from the sstem, namel, C and F. Similarl for the -determinant. Lewinter & Widulski The Saga of Mathematics 43 Lewinter & Widulski The Saga of Mathematics 44 Cramer s Rule The solution of the sstem is and provided of course that is not 0. Solve the following sstem using Cramer s rule: + 4 3 + 4 5 The three determinants, and are given on the net slide. Lewinter & Widulski The Saga of Mathematics 45 Lewinter & Widulski The Saga of Mathematics 46 (continued) (continued) 1 3 4 5 1 3 1 1 4 3 1 4 3 1 4 1 4 4 5 1 16 5 11 4 4 1 5 3 4 5 1 7 5 Since 0, the sstem has a solution, namel 11 11 1 and 7 7 1 Lewinter & Widulski The Saga of Mathematics 47 Lewinter & Widulski The Saga of Mathematics 48 Lewinter & Widulski 8
Algebraic Fractions In order to add (or subtract) algebraic fractions, ou need to find the LC. To find the LC: 1. Factor each denominator completel, using eponents to epress repeated factors.. Write the product of all the different factors. 3. For each factor, use the highest power of that factor in an of the denominators. To Add (or Subtract) Algebraic Fractions 1. Find the LC. Multipl each epression b missing factors missing factors 3. Add (or Subtract) 4. Simplif Lewinter & Widulski The Saga of Mathematics 49 Lewinter & Widulski The Saga of Mathematics 50 Solving Equations Involving Algebraic Fractions Add: 3 + LC ( )( + ) + 3( + ) 3( ) + ( )( + ) ( )( + ) 3( + ) + 3( ) 3 + 6 + 3 6 ( )( + ) ( )( + ) 6 3 ( )( + ) Multipl both sides of the equation b the LC. Be sure to use the distributive propert when necessar! Solve the resulting equation after simplifing both sides. Lewinter & Widulski The Saga of Mathematics 51 Lewinter & Widulski The Saga of Mathematics 5 (continued) 6 Solve: 5 11 + 5 6 5 6 5 5 + 5 6 ( 5) + 6( 5) ( 5) 6( 5) 6( 5) 11 6 11 6 ( ) + 6( 5)( 5) ( 5)( 11) LC 6 ( 5) 6 + 30 150 11 0 5 0 0 55 85 + 150 17 + 30 ( )( 15) or 15 Lewinter & Widulski The Saga of Mathematics 53 Lewinter & Widulski The Saga of Mathematics 54 Lewinter & Widulski 9
(continued) Be sure to check our answers and watch out for etraneous roots. Remember ou can not divide b zero, so if an answer results in a denominator having a value of zero that answer is an etraneous root. Let s look at an eample! Solve: 3 1 3 + + Lewinter & Widulski The Saga of Mathematics 55 Lewinter & Widulski The Saga of Mathematics 56 Problem Solving (Work) Problem Solving (Uniform Motion) Neil needs 15 minutes to do the dishes, while Bob can do them in 0 minutes. How long will it take them if the work together? Let the time it takes them working together Neil can do 1/15 of the work in 1 minute. Bob can do 1/0 of the work in 1 minute. Together: 1 1 1 + 15 0 Lewinter & Widulski The Saga of Mathematics 57 Jud drove her empt trailer 300 miles to Saratoga to pick up a load of horses. When the trailer was finall loaded, her average speed was 10 mph less than when the trailer was empt. If the return trip took her 1 hour longer, what was her average speed with the trailer empt? Let the average speed Use the formula: distance rate time Lewinter & Widulski The Saga of Mathematics 58 Equation of a Circle To find the equation of a circle centered at (a, b) with radius R, we use Pthagorean theorem. It follows that ( a) + ( b) R. (, ) R (a, b) ( a) ( b) Lewinter & Widulski The Saga of Mathematics 59 Lewinter & Widulski 10