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MATH 30 WEEK 3 SESSION 3 LESSON MATH 30 C1 SESSION 3 LESSON 1 SESSION 3 LESSON READ: Chapter 1, Sections 5, 6, 7 KEY POINTS Section 1.5 i. Solve a quadratic equation by factoring. Page 144. ii. Solve a quadratic equation by the square root property. Page 147. iii. Solve a quadratic equation using the quadratic formula. Page 151. iv. The discriminant. Page 154. What you need to know is how to use the discriminant to determine the kinds of solutions a quadratic equation has. In the first two cases, the equation has solutions that are real numbers. In the third case, the solutions are not real numbers; these imaginary solutions are called complex conjugates. Don’t worry about complex conjugates; we’re not going to deal with them. v. Determining the most effective technique for solving a quadratic equation. Page 156. NOTE: Only a very small percentage of quadratic equations can be solved by factoring or by the square root property, but they can all be solved by the quadratic formula. MATH 30 C1 INSTRUCTOR: PIRADEE NGANRUANG SESSION 3 LESSON 2 vi. The Pythagorean Theorem. Page 158. SKIP: Completing the square. Pages 148-151. Example 2 b. Page 147. Example 7. Page 153. Examples 2 b. and 7 have complex conjugate solutions, that is, solutions containing the number i. i stands for “imaginaryâ€Â. By definition, , a value that can only be imaginary because negative numbers do not have real square roots. Mathematicians invented i in order to solve equations with negative discriminants [page 154]. Operations with imaginary numbers are covered in Section 1.4. We skipped it, as it has limited application to the work we want to focus on. You will not be required to solve equations with imaginary solutions. You need to understand table 1.2 on page 154, and there will be a further short discussion of imaginary numbers in Session 8, but no operations will be performed. None of the practice exercises for Session 1.5 involves imaginary numbers. Practice Exercises: 1, 2, 15, 67, 71, 77, 79, 83, 84, 85, 141, 142 Section 1.6 i. Solve polynomial equations. Page 167. ii. Solve radical equations. Page 169. BE CAREFUL! Check all proposed solutions in the original equation. Consider the following: MATH 30 C1 INSTRUCTOR: PIRADEE NGANRUANG SESSION 3 LESSON 3 Now check both solutions in the original equation: For x = 6: True. For x = 1: False. The principal square root of 4 is 2, not . [You may want to revisit principal square roots on page 34.] iii. Solve equations with rational exponents. Page 172. Don’t be scared to check the solutions, even though it may involve raising a fraction to a fraction. Your calculator does this with ease, provided you are careful with the input. Consider Example 5 b. To check the first solution, , we substitute into the original equation: MATH 30 C1 INSTRUCTOR: PIRADEE NGANRUANG SESSION 3 LESSON 4 To evaluate the first term on your calculator, the input is: (-1/8)^(2/3). You must put brackets around both exponents. Input correctly, the answer is .25 or . We can then verify that the solution is correct: Make sure you can do this on your calculator. Then check the other solution as well. For anyone who is curious as to how can be evaluated without a calculator, we employ that useful definition: [In order to square , we used the Quotients-to-Powers Rule for Exponents, page 25, and to evaluate the cube root of , we used the Quotient Rule for nth Roots, page 43.] SKIP: Equations involving absolute value. Page 176. MATH 30 C1 INSTRUCTOR: PIRADEE NGANRUANG SESSION 3 LESSON 5 Practice Exercises: 1, 2, 11, 12, 14, 15, 32, 33, 105, 111 Section 1.7 i. Interval notation. Page 183. Also see Table 1.4 on page 184. You’re expected to be familiar with all the various ways that intervals can be expressed. For example, in describing an interval on the x-axis, you need to realize that: means exactly the same thing as: Remember, the arrow always points to the smaller number. ii. Solving a linear inequality. Page 187. It is just like solving an equation except that the solution contains an ordering symbol instead of the equals sign. iii. Properties of inequalities. Page 187. BE CAREFUL! Always remember the Negative Multiplication Property of Inequality. When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the arrow. iv. Compound inequalities. Page 191. The solution will contain two ordering symbols. SKIP: Solving inequalities with absolute value. Page 192. Practice Exercises: 27, 29, 30, 31, 37, 39, 52, 54, 119, 122, 125, 132
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MATH/30 MATH30 MATH 30 WEEK 3 SESSION 3 LESSON
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