Lesson #33 Solving Incomplete Quadratics

Transcription

1 Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~

2 We can also set up any quadratic to solve it in this way by completing the square, the technique we used to write circles in center-radius form. Review: Write y 6 4y 10 0 in center radius form. When we are solving a quadratic equation, the completing the square is actually a little easier because we only have to work with one variable. Eample: Solve the equation, Move the constant term to the other side of the equation. b b. Find and. Complete the square by adding b to both sides of the equation. 4. Factor the left side of the equation into a perfect square binomial. 5. Take the square root of both sides. Do not forget the plus or minus. 6. Add or subtract to solve for b b ( )( ) 17 ( ) You can use this process to solve any quadratic, but it is especially useful when the quadratic cannot be factored. ~ ~

3 A couple of special cases When b is odd, you will have to work with fractions. c) 5 0 d) When a is greater than 1, move c to the other side of the equation, then divide both sides by a and complete the square. e) f) 1 0 ~ ~

4 Lesson #4 The Quadratic Formula A.A.5 Solve quadratic equations, using the quadratic formula Simplify each of the following epressions: 11 6 = 50 6 = 7 6 = 5 6 = ~ 4 ~

5 The equation above can also so solved by finding the -intercepts of the function, f ( ) We are finding the roots or where f()=0. You already learned this last unit when we solved equations graphically in Lesson #0 and when we found the roots of higher order equations in Lesson #1. Now we are just connecting it to the quadratic formula. Eercise #4 continued c) Graph as Y and Y=0 and sketch the graph on the aes provided. Find the intersection of Y1 and Y, and label these points. How do your intersections compare to your answers to parts a and b? ~ 5 ~

6 ~ 6 ~ If one root of a quadratic equation is an irrational binomial, the other root will be its conjugate. Conjugates are numbers in the form a+b and a-b.

7 Lesson #5 The Sum and Product of the Roots A.A.5 Solve quadratic equations, using the quadratic formula ~ 7 ~

8 The net type of problem gives the sum and the product and asks you to find the equation. You can work backwards using the formulas we learned. 1. If the sum of the roots of a quadratic equation is 10 and the product of the roots is 9, write one possible equation with these roots. This is what we are looking for: 0 y Note: The problem tells us to write one possible equation. All other possible equations would be multiples of the given equation. If a=, the equation would be. First write both the sum and product as fractions. Make sure the fractions have the same denominator. 10 Sum: Product: 1 1 Net, match up the numbers with the variables in the formula. Sum: 10 1 Product: 9 1 b a c a Therefore, a=1, b= -10, and c=9 If a=, the equation would be. Your equation would be: If the sum of the roots of a quadratic equation is 7 and the product of the roots is 5, write one possible equation with these roots. (Since the denominator of the sum is, write 5 as 10.) ~ 8 ~

9 4. For which equation is the sum of the roots equal to the product of the roots? (1) 1 0 () () 6 0 (4) If the sum of the roots of a quadratic equation is 7 is, write one possible equation with these roots. and the product of the roots 7. If the sum of the roots of result is (1) 15 () () 15 (4) is added to the product of its roots, the 8. If the sum of the roots of a quadratic equation is 4 and the product of the roots is 8, write one possible equation with these roots. ~ 9 ~

10 A.N.6 A.N.7 Lesson #6 Imaginary Numbers Write square roots of negative numbers in terms of i Simplify powers of i If you factor -1 out of any negative number, you will be left with a number. For eample, 19 1 Solve the following quadratic using the quadratic formula: 7 0 Graph the quadratic above on your calculator. Where are the roots (zeroes)? What is wrong with the solution to the quadratic equation above? We could factor out of that number to make it positive. Now the 1 is the only problem with the answer. We use the letter i, known as the imaginary unit, to take the place of the 1. Therefore, 19 i 1 Any single term that contains i is called an. a. 4 = b. 16 = c. 100 = d. 4 9 e. 0 = f. = g. 4 7 = ~ 10 ~

11 You can think of i as a place holder for the 1 so that you can still work with negative radicals. Q: If negative radicals are not in the real number system, why would we want to work with them at all? A: All is not lost. Powers of i can become real numbers again. Powers of i i i i Simplified imaginary numbers have no eponent on i. Fill in the table: i i i 0 = i 4 = i 1 = i 5 = i = i 6 = i = i 7 = i 8 = i 1 = i 9 = i 1 = i 10 = i 14 = i 11 = i 15 = i 1 1 i = Without doing any work, what would you epect i 8 to equal? Fill in the rest of the table above. Describe the pattern that emerges Powers of i that have: Remainder 0 when divided by 4 ( ): 4 i = 5 i = 6 i = 7 i = Remainder 1 when divided by 4 ( ): Remainder when divided 4 ( ): Remainder when divided by 4 ( ): This type of pattern is cyclic because it repeats itself. Simplify: i R i i i You can do powers of i on your calculator too! Sometimes your calculator will make a rounding error that results in scientific notation for a very small number which is really zero. 65 i 1.5E 1 i i 19 i i i i ~ 11 ~ 4 5i

12 Write in simplest form: a. i 0 = b. i 5 = c. i 0 = d. i 4 = e. 5i 47 = f. -4i 10 = Eponent Rules and Powers of i When you work with i, it acts just like an or any other variable EXCEPT when you have an eponent on i. Operation Eample with Eample with i Multiplication 1.( the eponents) 0 6 = 0 6 i i = Division ( the eponents) 10 8 = 8 i i 10 = Power to a Power ( the eponents) ( ) = ( i ) = Simplify: 1) 10 5 i ( i ) = 9i i 15 ) 1 = ) i = ~ 1 ~

13 Lesson #7 Comple Numbers A.N.8 Determine the conjugate of a comple number A.N.9 Perform arithmetic operations on comple numbers and write the answer in the form a+ bi. Note: This includes simplifying epressions with comple denominators From last lesson you will remember that i is just a place holder for 1. When you work with i, it acts just like an or any other variable EXCEPT when you have a power of i. In that case you need to write the power of i without an eponent. The most common power of i that arises in these problems is i with a -1. which can always be replaced YOU MUST TAKE THE NEGATIVE OUT OF THE RADICAL AND MAKE IT AN i BEFORE DOING ANYTHING ELSE! Adding/ Subtracting Radicals Operations with Imaginary Numbers 48 7 Adding radicals 16 9 with i i i 9 4 4i i i Multiplying Radicals (10) 40 Dividing Radicals Multiplying radicals with i Dividing Radicals with i Remember, i = 1. Therefore you cannot have i in the denominator of a radical i 6 i 50 4 i 00 4( 1) 100 4(10) i 4 i i i 4i i 4i ~ 1 ~

14 1) When epressed as a monomial in terms of i, 5 8 is equivalent to (1) i () i () i (4) 18i ) Find the following product: 4 0 ) Simplify by rationalizing the denominator: 5 5 Comple numbers are any number in the form a+bi where a and b are both real numbers. a is the real component and bi is the imaginary component. Operations with Comple Numbers: Note the similarities and differences when working with a normal variable () and the imaginary unit, i. Operation With With i Adding (5+4)+(6+)= (5+4i)+(6+i)= Subtracting Multiplying 11+7 (9-)-(6-)= = + ( )( ) i (9-i)-(6-i)= 9-i-6+i = +i ( i)( i) 6 4i 9i 6i 6 5 i ( 6)( 1) 1 5i This is known as simplest a+bi form. You would not write the answer as 7i+11 because the imaginary component always comes second. 4) (7-4i)-(9-1i)= 5) What is the sum of 4 and 16 epressed in a + bi form? (1) 1 i () 1 1i () 1 i 0 (4) 14 i 6) (6 18) (4 50) = ~ 14 ~ Good News Most comple number problems can be simply typed in your calculator. Using the MODE button, set your calculator in a+bi mode. It is still important that you know what you are doing though.

15 We use this property of comple conjugates to simplify epressions with comple numbers in the denominator. This is the same eact process as what we did in Lesson #7 with irrational binomials in the denominator. (Remember, you cannot have i in the denominator because it is a radical.) 5 From lesson #7: Rationalize the denominator: 7 Epress the following quotients in simplest a+bi form. 5i 1 i 1. i. i ~ 15 ~ Good News Once again, these problems can also be typed in your calculator to check them. The only etra step you might have to do is changing the decimals to fractions.

16 Simplify each of the following epressions. 7) 7 8 5i i 4i i 8) The epression i is equivalent to i (1) 6 5 i () 7 5 i 8 10 () 6 i (4) 7 i ) 1 1 7i 4i 4i 10) What is the sum of 5 i and the conjugate of i? (1) + 5i () 8 + 5i () 5i (4) 8 5i 4 11) i (1 4i 8i i ) = 1) (1 i ) = 1) Impedance measures the opposition of an electrical circuit to the flow of electricity. The total Z1Z impedance in a particular circuit is given by the formula ZT. What is the total impedance of Z1 Z a circuit, Z T, if Z1 1 i and Z 1 i? (1) 1 () 5 () 0 (4) ~ 16 ~

17 Graphing Comple Numbers Real numbers can be graphed on a number line. For eample, graph = and =-1 on the number line provided. Comple numbers can also be graphed, but we will need another line for the imaginary component of the number. To do so we draw a vertical number line through the horizontal real number line. See the eample below. Sometimes, vectors or arrows are drawn from the origin to the point, but this is not necessary. 1. Draw a vector to each of the points on the aes provided.. Graph 5-4i.. Calculate ( i) on the calculator and graph the product. What is the sum of w and u, epressed in standard comple number form? (1) 7 + i () 5 + 7i () + 7i (4) -5 + i On a stamp honoring the German mathematician Carl Gauss, several comple numbers appear. The accompanying graph shows two of these numbers. Epress the sum of these numbers in a bi form. ~ 17 ~

18 Lesson #8 Equations with Comple Solutions A.A.5 Solve quadratic equations, using the quadratic formula A.N.6 Write square roots of negative numbers in terms of i A.A. Solve systems of equations involving one linear equation and one quadratic equation algebraically. Note: This includes rational equations that result in linear equations with etraneous roots earlier this unit Note: You can use either the quadratic formula or completing the square. Try a different method for each problem. ~ 18 ~

19 Solve each of the following incomplete quadratics. A. 9 0 B C. ( 4) 18 Solve the following system of equations: y y 1. Graph each of the above functions. Where do they intersect? ~ 19 ~

20 Lesson #9 The Discriminant A.A. Use the discriminant to determine the nature of the roots of a quadratic equation Decide if each set of numbers is: Equal or Unequal Real or Imaginary (Comple) If Real, rational or irrational a) 7, 7 b) 5 5, c) 0, 0 d) 9, 9 When do you usually see numbers in the form above? What part of each number dictates what type of number it is? b 4 ac is such an important part of the quadratic formula that it has its own name, the discriminant. If I were the math queen of the world, I would rename it the preview because it tells us what the roots will be like or their nature without having to take the time to find them. The key to working with the discriminant is to remember where it came from: the radical in the quadratic formula. Therefore you can always tell what the roots will be like if you ask yourself the question, what would happen if that number was in a? a) Use the discriminant to describe the nature of the roots of the quadratic equation, b) Use the discriminant to describe the nature of the roots of the quadratic equation, 4 0. c) Use the discriminant to describe the nature of the roots of the quadratic equation, ~ 0 ~

21 d) Use the discriminant to describe the nature of the roots of the quadratic equation, 5 0. Solve the Equation using the quadratic formula ( 7) ( 7) 4()( 6) () Complete the table below as part of your Lesson #9 homework Nature of the Roots (Type of numbers) Choices: (Equal or Unequal) (Real or Imaginary) (If Real, rational or irrational) Unequal Real Rational 4 0 (Equal or Unequal) (Real or Imaginary) (If Real, rational or irrational) Rough Sketch from your calculator (Equal or Unequal) (Real or Imaginary) (If Real, rational or irrational) 5 0 (Equal or Unequal) (Real or Imaginary) (If Real, rational or irrational) ~ 1 ~

22 Compare your answers to a-d on page 19-0 with the information in the table on page 0. The discriminant can quickly tell you what the roots will be like which also tells you how many times the graph of the quadratic will cross the -ais. Which of your two sketches look the most similar in terms of their zeroes? Describe what the graph with imaginary roots looks like. Describe what the graph with one, equal root looks like. Equation Discriminant Summary Table Discriminant? b 4ac Nature of Roots ( 7) 4()( 6) 11 # of times graph intersects -ais (zeroes) 4 0 Positive, a perfect square ( ) 4()(4) A negative number ( 6) 4(1)(9) Zero 0 ( 5) 4(1)( ) Positive, not a perfect square NOTE: THE DISCRIMINANT IS NOT THE ROOTS. It tells you what the roots will be like, but not the actual value. Think of it like a movie preview. From a preview (discriminant), you know what type of movie you are going to (irrational, rational, imaginary, etc.), but you have to actually watch the movie (solve the equation) to know all of the details (solutions). ~ ~

23 1. Use the discriminant to find the nature of the roots.. Based upon the nature of the roots, state whether the graph of the quadratic intersects the -ais twice, once, or never. a) 4 b) c) d) There are many different types of questions that can be asked about the roots of a quadratic equation. While the discriminant is the quickest way to solve these problems, some can also be solved with a graph or the entire quadratic formula. 1 The roots of a quadratic equation are real, rational, and equal when the discriminant is (1) () 0 () (4) 4 Which number is the discriminant of a quadratic equation whose roots are real, unequal, and irrational? (1) 0 () 7 () -5 (4) b Jacob is solving a quadratic equation. He eecutes a program on his graphing calculator and sees that the roots are real, rational, and unequal. This information indicates to Jacob that the discriminant is (1) zero () a perfect square () negative (4) not a perfect square ~ ~

24 b b 6 The roots of the equation 0 are (1) real, rational, and equal () real, rational, and unequal () real, irrational, and unequal (4) imaginary The roots of the equation are (1) imaginary () real, rational, and equal () real, irrational, and unequal (4) real, rational, and unequal What type of numbers are the roots of the equation, 5 1 0? b 7 What type of numbers are the roots of the equation,? 06019b b b b The roots of the equation 5 0 are (1) imaginary () real, rational, and equal () real, rational, and unequal (4) real and irrational Which equation has roots that are real, rational, and unequal? (1) 1 0 () 4 0 () (4) 0 Which equation has imaginary roots? (1) 1 0 () 1 0 () 0 (4) 1 0 ~ 4 ~

25 b The equation 8 n 0 has imaginary roots when n is equal to (1) 10 () 6 () 8 (4) b b b Which statement must be true if a parabola represented by the equation y a b c does not intersect the -ais? (1) b 4ac 0 () b 4ac 0 () b 4ac 0, and b 4 ac is a perfect square. (4) b 4ac 0, and b 4 ac is not a perfect square. If the roots of a b c 0 are real, rational, and equal, what is true about the graph of the function y a b c? (1) It intersects the -ais in two distinct points. () It lies entirely below the -ais. () It lies entirely above the -ais. (4) It is tangent to the -ais. Which is a true statement about the graph of the equation y 7 60? (1) It is tangent to the -ais. () It does not intersect the -ais. () It intersects the -ais in two distinct points that have irrational coordinates. (4) It intersects the -ais in two distinct points that have rational coordinates b Which graph represents a quadratic function with a negative discriminant? ~ 5 ~

26 Lesson #40 Solving Higher Order Equations A.A.6 Find the solution to polynomial equations of higher degrees that can be solved using factoring and/or the quadratic formula Higher order polynomial equations are those where the variable is cubed, to the fourth power, or higher. An eample would be. Some of these equations, including the one in this paragraph, require advanced factoring techniques to solve. The ones we will be working with in this course can all be solved using the techniques you already know for quadratics including factoring, the quadratic formula, and incomplete quadratics. What is the maimum number of solutions for a linear equation? quadratic equation? cubic equation? How can you generalize the maimum number of solutions to a polynomial equation? The following three situations summarize the different methods we will be using. 1. With a GCF (we have already done some like this) ( 4 ) 0 5 ( )( 1) or 0 or 0 or or 0 or or 1, 1,0 Notes Solve for zero if necessary. Factor out the GCF. Factor again if possible. Set each factor equal to zero and solve. If any of your new equations is a quadratic use the quadratic formula or the method for incomplete quadratics to solve that part of the equation rest of the equation.. In the form: 4 a b c ( ) 1( ) 6 0 ( 9)( 4) or ,,, Notes Solve for zero if necessary. Write 4 as. Factor the equation as if was really. ( in this case.) Set each factor equal to zero and solve. You can usually use the method we learned for incomplete quadratics. ~ 6 ~

27 . Factoring by Grouping Notes This method is used with a cubic that does not have a GCF. 5 ( ) ( ) 0 (5 )( ) or 0 5 or Factor by grouping Set each factor equal to zero. Solve each resulting equation using an appropriate method ,, 5 5 Note: Solving polynomials of higher degree takes practice and some creativity to find the best method. As long as the solutions are not imaginary, you could also use a graph to find the roots like we learned last unit. The only problem is that irrational solutions will be estimated on the graph, not in radical form. Look at 1. Where would the graph of 5 4 y intersect the -ais? Look at. Where would the graph of 4 y 1 6 intersect the -ais? Look at. Where would the graph of nearest tenth. y intersect the -ais? Round to the Solve each of the following equations t 10t 1t 0. a a a 0 ~ 7 ~

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29 Lesson #41 - More About Completing the Square Review Problems 1) For the equation, a) Use the discriminant to determine the nature of the roots. b) Find the sum and the product of the roots. c) Solve the equation by completing the square. d) Solve the equation using the quadratic formula. e) Add and multiply the roots together to see if you get the same answers as b. B is odd. Solve the following equations: 4 ) ) ~ 9 ~

30 In lesson # you learned how to complete the square in order to solve quadratic equations. All of the eamples involved equations where the value of a 1or a was a common factor of each term. In each of the following problems, a 1 and will not divide out evenly A B D C E F. G , 4 4 a is not equal to 1. b Move the constant term to the other side of the equation. 8. Divide both sides of the equation by a. b b 9. Find and. 10. Complete the square by adding to both sides of the equation. 11. Factor the left side of the equation into a perfect square binomial. 1. Take the square root of both sides. Do not forget the plus or minus. 1. Add or subtract to solve for. b 1) 4 0 ) 5 0 ~ 0 ~

31 The General Case Since the process for the completing the square is always the same, we can solve for in the equation, a b c 0to generalize the process. You can use the eample on the left if you have difficulty completing the process with variables b a b c ~ 1 ~