Unit #8 Radicals and the Quadratic Formula


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1 Unit #8 Radicals and the Quadratic Formula Lessons: 1  Square Root Functions 2  Solving Square Root Equations  The Basic Exponent Properties 4  Fractional Exponents Revisited 5  More Exponent Practice 6  The Quadratic Formula 7  More Work with the Quadratic Formula
2 Lesson 1: Square Root Functions and Graphs Square roots are the natural inverses of squaring. In other words, to find the square root of an input, we must find a number that when squared gives the input. Because of their important role in higherlevel mathematics, it is important to understand their graphs, as well as their domains and ranges. f x x and g x x 2. Exercise #1: Consider the two functions (a) Graph y = f(x) without the use of your calculator on the grid shown. Label its equation. y (b) Using your calculator to generate a table of values, graph y = g(x) on the same grid and label its equation. Start your table at x = 10 to see certain xvalues not in the domain of this function. x (c) State the domain and range of each function below using setbuilder notation. f(x) = x g(x) = x + 2 Domain = Domain = Range = Range = Shifts in Graphs: We ve seen in previous chapter that horizontal movement (left or right) is controlled by any constants inside with the xterm, and that vertical movement (up or down) is controlled by any constants grouped with the yterm or alone. Additionally, coefficients effect the graph by changing how wide or narrow the graph is, and can change the direction that a graph goes in. For example, y = x 2 would open downwards because of the negative coefficient. These major shifts hold true for many function graphs, including the square root y Exercise #2: Which of the following equations would represent the graph shown? (1) y = x + 4 (2) y = 4 x () y = x 4 (4) y = x 4 x
3 As we saw in the first exercise, the domains of square root functions are oftentimes limited due to the fact that square roots of negative numbers do not exist in the Real Number System. We shall see in Unit #9 how these square roots can be defined if a new type of number is introduced. For now, though, we are only working with real numbers. Recall: To determine the domain for these functions, we take the radicand (term(s) under the square root) and set it greater than or equal to zero, and solve. radicand 0 Exercise #: Which of the following values of x does not lie in the domain of the function y = x 5? Explain why it does not lie there. (1) x = 6 () x = 5 (2) x = 2 (4) x = 7 Exercise #4: Determine the domain for each of the following functions. Show an inequality that justifies you work. (a) y = x + 2 (b) y = x 2 (c) y = 8 2x Exercise #5: Consider the function f(x) = x 2 + 4x 12. y (a) Use your calculator to sketch the function on the axes given. x (b) Set up and solve a quadratic inequality that yields the domain of f(x).
4 Lesson 1 Homework: Square Root Functions and Graphs 1. Which of the following represents the domain and range of y = x 5 + 7? (1) Domain: [ 5, ) () Domain: ( 7, ) Range: [7, ) Range: (5, ) (2) Domain: [5, ) (4) Domain: [7, ) Range: [7, ) Range: [5, ) 2. Which of the following values of x is not in the domain of y = 1 x? (1) x = 1 () x = 0 (2) x = 1 (4) x = 4. Which of the following equations 4. Which equation below represents the describes the graph shown below? graph shown below? y (1) y = x (1) y = x 2 5 y (2) y = x 4 1 (2) y = x () y = x () y = x x x (4) y = x (4) y = x Determine the domains of each of the following functions. State your answers in setbuilder notation. (a) y = x + 10 (b) y = x 5 (c) y = 7 2x
5 6. Set up and algebraically solve a quadratic inequality that results in the domain of each of the following. Verify your answers by graphing the function in a standard viewing window. (a) y = x 2 4x 5 (b) y = 9 x 2 7. Consider the function g(x) = x (a) Graph the function y g x on the grid shown. y (b) Describe the transformations that have occurred to the graph of y x to produce the graph of y g x. Specify both the transformations and their order. x
6 Answers to Lesson 1 Homework 1) (2) 2) (4) ) (1) 4) () 5) (a) {x 10} (b) {x 5 } (c) {x 7 } 2 6) (a) (b) First the function is reflected in the xaxis. Then, the function is shifted 5 units to the left and shifted units up.
7 Lesson 2: Solving Square Root Functions Equations involving square roots arise in a variety of contexts, both applied and purely mathematical. As always, the key to solving these equations lies in the applications of inverse operations. Recall that a square and square root are inverses. Get the square root alone Square both sides Finish solving Check all answers for extraneous roots Oftentimes, roots are introduced by various algebraic techniques that for one reason or another are not valid solutions of the equations. These roots are known as extraneous and can always be found by calculatorchecking within the original equation. Exercise #1: Solve each of the following square root equations. Check each equation for extraneous roots (a) x = 7 (b) x = 5 (c) 2x 1 = 4 (d) x 4 = 20 (e) 2 x = 1 (f) 5 x 2 4 = 6 Exercise #2: Which of the following is the solution to x 2 = 15? (1) x = 12.5 (2) x = 50 () x = 25 (4) x = 4050
8 Another scenario arises when a square root expression is equal to a linear expression. The next exercise will illustrate both the graphical and algebraic issues involved. y Exercise #: Consider the system of equations shown below. y = x + and y = x + 1 (a) Solve this system graphically using the grid to the right. x (b) Solve this system algebraically for only the xvalues using substitution below. (c) Why does your answer from part (a) contradict what you found in part (b)? Exercise #4: Find the solution set of each of the following. Be sure to check your work and reject any extraneous roots. (a) 2x = x (b) 2x = x + 6 2
9 Lesson 2 Homework: Solving Square Root Functions 1. Solve and check each of the following equations. As in the lesson, they are arranged from lesser to more complex. (a) x = 5 (b) x + 2 = 10 (c) 2x = 6 (d) 4 x = 24 (e) 2 x = 1 (f) x = 4 = 8 (g) 1 x 5 = 2 (h) 4x 1 + = 4 (i) 5 1 5x = 27 2
10 2. Which of the following values solves the equation 4x+19 = 2? 2 (1) 9 (2) 2 4 () 4 (4) 1 2. Solve and check each of the following equations for all values of x. Reject any extraneous roots. (a) x 1 = x + 11 (b) 4x + 6 = 2x 6. (c) 6x = 2 2x (d) 6x = x
11 Answers to Lesson 2 Homework 1) (a) x = 25 (b) x = 98 (c) x = 54 (d) x = 6 (e) x = 1 4 (f) x = 20 (g) x = 196 (h) x = 1 2 (i) x = 7 2) (2) ) (a) x = 5; x 2 (extraneous root) (b) x = 7; x 0 (extraneous root) 4) (a) x = {± 1 } (b) x = ; x 1 (extraneous root) 8 2
12 Lesson : The Basic Exponent Properties Exponents, which indicate repeated multiplication, are extremely important in higherlevel mathematical study because of their importance in numerous areas. The rules they play by, known as the exponent properties, are critical to master. You should have seen these properties in previous math courses. Recall Exponent Properties or Exponent Laws: 1) (x a )(x b ) = x a+b 2) xa x b xa b ; where x 0 ) (x a ) b = x a b 4) (x y) a = x a y a and ( x y )a = xa y a 5) x a = 1 1 and = xa 6) xa x a x 0 = 1 Exercise #1 (Property #1): Rewrite each of the following in simplest form: (a) x 10 x = (b) (5x 4 )(6x ) = (c) x y 2 x 6 y = Exercise #2 (Property #2): Rewrite each of the following in simplest form: (a) x8 x 2 = 6x10 (b) = (c) 2x6 y = 12x 4 8xy 2 Exercise # (Property #): Simplify each of the following: (a) (x 2 ) = (b) (y 4 ) 6 = (c) (a 1 6 2)
13 (d) Which of the following expressions is not equivalent to (1) (x 10 ) () x 5 x 6 (2) (x 6 ) 5 (4) x 10 x 20 0 x? Exercise #4 (Property #4): Rewrite each of the following as equivalent expressions: (a) (2x 2 ) = (b) ( x 2)4 = (c) ( 2x2 y 5 z ) = Exercise #5 (Property #5): Rewrite each of the following without the use of negative exponents: (a) 2 (b) 4 x (c) 1 (d) x4 y 10 x 15x 6 y 9 (e) 5x4 y 5x 2 y 8 (f) (x10 y 5 x 2 y ) 2 Handle the negatives Reduce the inside as much as possible Raise to the outside power Exercise #6 (Property #6): Simplify each of the following: (a) (b) x (c) 4(2x) 0 =
14 Lesson Homework: The Basic Exponent Properties 1. Express each of the following expressions in "expanded" form, i.e., do all of the multiplication and/or division possible and combine as many exponents as possible. (a) x x 12 (b) 4x 5x 5 (c) ( x 2 y)(5x 7 y ) (d) (4x y 6 )( 7x 4 ) (e) x9 (f) 5x y 7 x 15xy 2 x (g) (h) 10x4 y x 10 25x 8 (i) (x 5 ) 8 (j) (10x ) 0 (k) ( 4x 5 ) (l) (2x 2 ) 4 2. Which of the following is not equal to 2 2? Do not use your calculator to do this problem. Show your algebra. (1) 1 4 () 0.25 (2) 4 (4) If the expression 1 2x was placed in the form axb where a and b are real numbers, then which of the following is equal to a + b? Show how you arrived at your answer. (1) 1 () 1 2 (2) 2 (4) 1 2
15 4. If f(x) = 5x 0 + 4x then f(a) = (1) 12a 5 () 1 4a + 5 (2) (4) 12a + 1 a 5. Which of the following is equivalent to (4x8 ) (6x 5 )2 for all x 0? Show the work that leads to your final answer. (1) 16 9 x14 () 2 x14 (2) 16 9 x4 (4) 2 x4
16 Answers to Lesson Homework 1) (a) x 15 (b) 20x 8 (c) 15x 9 y 4 (d) 28x 7 y 6 (e) x 6 (f) 1 x2 y 5 (g) 1 2y (h) x7 5x 4 (i) x 40 (j) 1 (k) 64x 15 (l) 16 x 8 2) (2) Algebra required ) (4) Algebra required 4) (2) 5) (1)
17 Lesson 4: Fractional Exponents Revisited Recall that in Unit #4 we introduced the concept that roots could be represented by rational or fractional exponents. UNIT FRACTION EXPONENTS For n given as a positive integer: a 1 n n = a Recall we can also combine integer powers with roots with the following: power root power over root RATIONAL EXPONENT CONNECTION TO ROOTS For the rational number p, r ap r r is equivalent to: a p r or ( a) p Exercise #1: Rewrite each expression in the form (a) 5 x 5 (b) x 4 b ax where a and b are both rational numbers. (c) 7 (d) x 5 10 x Exercise #2: Rewrite each of the following power/root combinations as a rational exponent in simplest form. (a) x 7 4 (b) x 6 (c) ( x) 6 (d) ( x) 10 Exercise #: If f(x) = 10x 2 24x 1, then which of the following represents the value of f(4)? Find the value without the use of a calculator. Show the steps in your calculation. (1) 6 () 54 (2) 48 (4) 74 Exercise #4: Which of the following is not equivalent to x 7? (1) 1 () 1 x 7 x 7 1 (2) 7 (4) 1 x 7 x
18 Fractional exponents play by the same rules (properties) as all other exponents. It is, in fact, these properties that can justify many standard manipulations with square roots (and others). For example, simplifying roots. SIMPLIFYING RADICALS n a b = a b or a b n = a n b; try to keep a as the largest perfect square divisor Exercise #5: Simplify each of the following roots. Show manipulations. Be thoughtful about the index being used! It may be helpful to write or calculate a list of perfect squares, cubes, etc (a) 28 (b) x 6 y 11 Variables are perfect roots if the exponent is divisible by the index (c) 18x 4 (d) 200x 5 y (e) 147x 9 y 4 (f) 16 (g) 108 (h) 250 (i) 12x 8 4 (j) (k) 16x 8 4 (l) 48x 10 y 5 5 (m) 64x 12 y 15
19 Lesson 4 Homework: Fractional Exponents Revisited 1. Which of the following is equivalent to x 5 2? (1) 5x 2 (2) 2x 5 () x 5 5 (4) x 2 2. If the expression 1 x was placed in xa form, then which of the following would be the value of a? (1) 2 (2) 2 () 1 2 (4) 1 2. Which of the following is not equivalent to x 9? (1) x (2) ( x) 9 () x 9 2 (4) x 4 x 4. The radical expression 50x 5 y can be rewritten equivalently as (1) 25xy 2xy () 5x 2 y 2xy (2) 5xy xy (4) 10x 2 y 5xy 5. If the function y = 12 x was placed in the form y = ab x then which of the following is the value of a b? (1) 6 () 6 (2) 4 (4) 4 6. Rewrite each of the following expressions without roots by using fractional exponents. (a) x 5 (b) x 7 (c) x (d) x 6 (e) x 11 (f) (g) (h) 1 x x 2 x 9
20 7. Rewrite each of the following without the use of fractional or negative exponents by using radicals. (a) x 1 6 (b) x 1 10 (c) x 1 (d) x 1 5 (e) x 5 (f) x 7 2 (g) x 9 4 (h) x Simplify each of the following square roots that contain variables in the radicand. (a) 8x 9 (b) 75x 16 y 11 (c) 2x 18x 7 (d) x 2 y 98x 5 y 8 9. Express each of the following roots in simplest radical form. (a) 16x 8 (b) 108x 5 y 10 (c) 64x 12 y 14 (d) 75x 7 y Mikayla was trying to rewrite the expression 25x 1 2 in an equivalent form that is more convenient to use. She incorrectly rewrote it as 5 x. Explain Mikalya's error.
21 Answers to Lesson 4 Homework 1) () 2) (4) ) (1) 4) () 5) (4) 6) (a) x 1 2 (b) x 1 (c) x 1 7 (d) x 5 2 (e) x 11 (f) x 1 4 (g) x 2 (h) x ) (a) x 10 (b) x (c) 1 (d) x 1 5 x 5 (e) x (f) 1 x 7 4 (g) x9 (h) 1 11 x 2 8) (a) 2x 4 2x (b) 5x 8 y 5 y (c) 6x 4 2x (d) 21x 4 y 5 2x 9) (a) 2x 2x 2 (b) xy 4x 2 y (c) 4x 4 y 4 y 2 (d) 5x 2 y xy 2 10) Mikalya raised both the x and the 25 to the 1 2 power. However, only the x should be raised to the 1 2 power. The correct answer should be 25 x. She would be correct if the original expression contained parentheses: (25x) 1 2.
22 Lesson 5: Exponent Practice For further study in mathematics, especially Calculus, it is important to be able to manipulate expressions involving exponents, whether those exponents are positive, negative, or fractional. The basic laws of exponents, which you should have learned in Algebra 1 and have used previously in this course, are shown to the right. They apply regardless of the nature of the exponent (i.e. positive, negative, or fractional). EXPONENT LAWS 1. x a x b = x a+b 5. x a = 1 1 and xa x 2. xa = xa b xb 6. (x a ) b = x a b. (x y) a = x a y a and ( x y )a = xa 4. x m n n = x m (For integers m and n) y a 7. x 0 = 1 a = xa Make each step carefully & thoughtfully, keeping in mind the order in which you are working. Exercise #1: Simplify each of the following expressions. Leave no negative exponents in your answers. (a) x x 4 (x 5 )2 (b) (x2 4 y) x 5 y 7 (c) x y 4 x 6 y (d) (x y 4 ) 2 (xy ) 4 In the last exercise, all of the powers were integers. In the next exercise, we introduce fractional powers. Remember, though, that they will still follow the exponent rules above. If needed, use your calculator to help add and subtract the powers. Exercise #2: Simplify each of the following expressions. Write each without the use of negative exponents. (a) x1 x 1 2 x 1 6 (b) (x 1 2 ) 5 x 2 x (c) (4x 2 ) 2x 8
23 To be fully simplified, an expression should not contain negative exponents and should not contain fraction exponents. Exercise #: Rewrite each expression below in its simplest form (a) x 5 (b) x5 2 (c) 1 x 4 x 2 (d) x x (e) (8x 5 ) 2 (f) (27x)1 6 x Consider the real number component and the variable component individually. Be thoughtful about what is actually happening to which piece: Exercise #4: Which of the following is equivalent to 8x 7? (1) 8x 7 () 2x 7 (2) 2x 7 (4) 8x 7 Exercise #5: The expression (1) 1 2 x 1 2 () 4x x is the same as (2) 2x 1 2 (4) 1 2 x1 2
24 Lesson 5 Homework: Exponent Practice 1. Rewrite each of the following expressions in simplest form and without negative exponents. (a) x x 7 (x 2 ) (b) 5x 4 (c) (x y 4 ) (d) (2x ) 25x 10 (x y) 2 8x Which of the following represents the value of a 4 when a = and b = 2? 2 b (1) 4 9 (2) 4 81 () 1 6 (4) 1. Simplify each expression below so that it contains no negative exponents. Do not write the expressions using radicals. (a) x7 2 y 1 2 x 4 y 2 (b) (x 1 ) 4 x 2 (c) (5x2 y 1 2 ) (2x 2 y ) 4. Which of the following represents the expression 24 x 1 2 (1) 4 x2 () x 4 (2) 4x (4) 4x 2 6x 5 2 written in simplest form?
25 5. Rewrite each of the following expressions using radicals. Express your answers in simplest form. (a) (4x) 2 (b) x 2 (c) (x 4 ) 5 x 4 5 (d) x x (e) x x2 x 5 (f) (2 x) 24x 6. Which of the following is equivalent to 5 x 20x? 1 (1) () 1 4 x 5 4 x 2 4 (2) (4) 1 x 5 4 x 5 7. When written in terms of a fractional exponent the expression x x is x 2 (1) x 7 2 () x 1 2 (2) x 5 2 (4) x 2 8. Expressed as a radical expression, the fraction x1 x 1 2 is x 1 (1) () x 6 x 1 (2) 11 (4) x 11 x 6 6
26 Answers to Lesson 5 Homework 1) (a) x 4 (b) 1 5x 6 y5 (c) x (d) 4x18 2) (2) ) (a) x11 4 y 2 (b) x2 (c) 10x 8 y 7 2 4) (1) 5) (a) 8x x (b) 1 (c) x 2 x 2 x 2 5 (d) 1 6 (e) x 5 x 6 (f) x 6) (4) 7) (1) 8) (4)
27 Lesson 6: The Quadratic Formula There are three main paths to solving for the roots (solutions, zeros, xintercepts) of a trinomial equation: Factoring (although not all are factorable!) Completing the square Quadratic Formula Quadratic Formula: x = b± b2 4ac 2a Exercise #1: Solve the following quadratic using the given method. Express your answers in simplest radical form. (a) completing the square. (b) the quadratic formula. x 2 6x + 1 = 0 x 2 6x + 1 = 0 Notice that factoring would not work in this case. There are no numbers which multiply to +1 and add to 6! Exercise #2: Which of the following represents the solutions to the equation x 2 10x + 20 = 0? (1) x = {5 ± 10} () x = { 5 ± 10} (2) x = { 5 ± 5} (4) x = {5 ± 5} Notice that factoring would not work in this case. There are no numbers which multiply to +20 and add to 10! Make a conclusion: When a trinomial is NOT factorable, the roots will contain:
28 Exercise #: Solve the following quadratic using the given method. (a) factoring (b) the quadratic formula 2x x 6 = 0 2x x 6 = 0 Exercise #4: Solve each of the following quadratics by using the quadratic formula. Place all answers in simplest form. (a) x 2 + 5x + 2 = 0 (b) x 2 8x + 1 = 0 (c) 2x 2 2x 5 = 0 (d) 5x 2 + 8x 4 = 0
29 Lesson 6 Homework: The Quadratic Formula 1. Solve each of the following quadratic equations using the quadratic formula. Express all answers in simplest form. x = b ± b 2 4ac 2a (a) x 2 + 7x 18 = 0 (b) x 2 2x 1 = 0 (c) x 2 + 8x + 1 = 0 (d) x 2 2x = 0 (e) 6x 2 7x + 2 = 0 (f) 5x 2 + x 4 = 0 Remember the conclusion made at the end of exercise two above. Circle a big green circle around any trinomial that must have been factorable, based on your solutions.
30 2. Which of the following represents all solutions of x 2 4x 1 = 0? (1) {2 ± 5} () {2 ± 10} (2) {2 ± 5} (4) { 2 ± 12}. Which of the following is the solution set of the equation 4x 2 12x 19 = 0? (1) { 5 2 ± } () { 2 ± 7} (2) { 2 ± 2} (4) { 7 ± 6} 4. Rounded to the nearest hundredth the larger root of x 2 22x = 0 is (1) () 6.74 (2) 1.25 (4) 14.61
31 5. Algebraically find the xintercepts of the function y = x 2 4x 6. Express your answers in simplest radical form. 6. A missile is fired such that its height above the ground is given by h = 9.8t t + 6.5, where t represents the number of seconds since the rocket was fired. Using the quadratic formula, determine, to the nearest tenth of a second, when the rocket will hit the ground.
32 Answers to Lesson 6 Homework 1) (a) x = { 9, 2} (b) x = {1 ± 2} (c) x = { 4 ± } 2) (1) ) () 4) (4) (d) x = { 1± 10 } (e) x = { 1, 2 } (f) x = { ± 89} 2 10 (You should have circled two questions) 5) x = {2 ± 10} 6) t = { 8.2± } 19.6 t = { 0. 16, } Since the time must be positive, we reject the negative time. It will take approximately 4.1 seconds for the rocket to hit the ground.
33 Lesson 7: More Quadratic Formula Quadratic Formula: x = b± b 2 4ac 2a The quadratic formula (shown above) is extremely useful because it allows us to solve quadratic equations, whether they are prime or factorable. In this lesson, we will get more practice using this formula. Exercise #1: Consider the quadratic function f(x) = x 2 4x 6. (a) Algebraically determine this function s xintercepts using the quadratic formula. Write the answers in simplest radical form AND to the nearest hundredth 50 y x 20 (b) Find the vertex of this parabola (c) Sketch a graph of the quadratic on the axes given. Use the ZERO command on your calculator to verify your answers from part (b). Label the zeros on the graph.
34 Exercise #2: Which of the following sets represents the xintercepts of y = x 2 19x + 6? (1) { 1 2, 7 } (2) {2 5, 2 + 5} () { , } (4) { 1, 6} Exercise #: (Revisiting the Crazy Carmel Corn Company) Recall that the Crazy Carmel Corn company modeled the percent of popcorn kernels that would pop, P, as a function of the oil temperature, T, in degrees Fahrenheit using the equation P = T T 94 The company would like to find the range of temperatures that ensures that at least 50% of the kernels will pop. Write an inequality whose result is the temperature range the company would like to find. Solve this inequality with the help of the quadratic formula. Round all temperatures to the nearest tenth of a degree.
35 Exercise #4: Find the intersection points of the linearquadratic system shown below algebraically using the quadratic formula. Then, use you calculator to help produce a sketch of the system. Label the intersection points you found on your graph. y = 4x 2 6x + 2 and y = 6x y 15 5 x Note: The fact that the solutions to this system were rational numbers indicates that the quadratic equation in Exercise #4 could have been solved using factoring and the Zero Product Law
36 Lesson 7: More Quadratic Formula x = b ± b2 4ac 2a 1. Which of the following represents the solutions to x 2 4x + 12 = 6x 2? (1) x = {4 ± 7} () x = {5 ± 22} (2) x = {5 ± 11} (4) x = {4 ± 1} 2. The smaller root, to the nearest hundredth, of 2x 2 x 1 = 0 is (1) 0.28 () 1.78 (2) 0.50 (4).47. The xintercepts of y = 2x 2 + 7x 0 are (1) x = { 7± 191 } () x = { 6, 5 } 2 2 (2) x = {, 5} (4) x = { ± 11} Solve the following equation for all values of x. Express your answers in simplest radical form. 4) 4x 2 4x 5 = 8x + 6 5) 9x 2 = 6x + 4
37 6. Algebraically solve the system of equations shown below. Note that you can use either factoring or the quadratic formula to find the xcoordinates, but the quadratic formula is probably easier. y = 6x x 15 and y = 12x The Celsius temperature, C, of a chemical reaction increases and then decreases over time according to the formula C(t) = 1 2 t2 + 8t + 9, where t represents the time in minutes. Use the Quadratic Formula to help determine the amount of time, to the nearest tenth of a minute, it takes for the reaction to reach 110 degrees Celsius.
38 Answers to Lesson 7 Homework 1) (2) 2) (1) ) () 4) x = { ±2 5 } 2 5) x = { 1± 5 } 6) {( 6, 87), ( 5 6, 5)} 7) t = { 8± 0 } 1 t {2.5, 1.5} It will take approximately 2.5 minutes for the chemical reaction to reach 110 Celcius.
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