P.5 Solving Equations

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1 PRC Ch P_5.notebook P.5 Solving Equations What you should learn How to solve linear equations How to solve quadratic equations equations How to solve polynomial equations of degree three or higher How to solve equations involving radicals How to solve equations involving aboslute values Aug 5 5:50 PM Aug 5 6:01 PM 1

2 P.5 Solving Equations What you should learn What an equation is What types of solutions equations can have How to identify an equation as linear What equivalent equations are How to use equivalent equations to solve linear equations How to identify extraneous solutions Aug 5 5:50 PM Equations and Solutions of Equations Examples: An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means... to find all of the values of x for which the equation is true. These values are called solutions. For example: x = 4 is a solution to 3x 5 = 7 because 3(4) 5 = 7 Substitution Principle Aug 9 8:40 AM 2

3 Equations and Solutions of Equations The existence of solutions depends on... the types of numbers being considered. No solutions over the integers No solutions over the rationals Two solutions over the irrationals Two solutions over the reals. Two solutions over the complex. Aug 9 9:56 AM Equations and Solutions of Equations An identity is... For example: an equation that is true for every real number in the domain of the variable. Identity domain: R Identity domain: U Aug 9 9:56 AM 3

4 Equations and Solutions of Equations A conditional equation is... For example: an equation that is not true for every (or any) value in the domain of the variable. (Conditional Equation) domain: R The only solutions to this equation are. (Conditional Equation) domain: R Their are no real values of x for which this equation is true. Aug 9 9:56 AM Linear Equations A linear equation in one variable (x) is... an equation that can be written in the form: ax + b = 0 where a and b are real numbers and a = 0. A linear equation has exactly one solution. Consider: Aug 9 9:56 AM 4

5 Equivalent Equations An equivalent equation is... Recall from previous slide: an equation that has the same solution as the original equation. (Original Equation) (Equivalent Equation) (Equivalent Equation) To solve a conditional equation in x... isolate x on one side of the equation by a sequence of equivalent equations. Aug 9 9:56 AM Generating Equivalent Equations An equation can be transformed into a equivalent equation by one or more of the following steps. Given Equation Equivalent Equation Remove symbols of grouping, combine like terms, or simplify fractions on one or both sides. Add (or subtract) the same quantity to (or from) both sides. 3. Multiply (or divide) both sides by the same non zero quantity. 4. Swap both sides. Aug 9 9:56 AM 5

6 Example 1 Solving a Linear Equation Solve. Solution: Check: (Substitution Principle) Aug 9 11:34 AM Example 2 An Equation Involving Fractional Expressions Solve. Solution: Aug 9 11:34 AM 6

7 PRC Ch P_5.notebook Example 3 An Equation with an Extraneous Solution Solve. Solution: Aug 9 11:34 AM Aug 9 3:56 PM 7

8 SUMMARY: An equation is two expressions set equal to each other. Identity Equations are true for every value of the variable in the domain. Conditional Equations are not true for all or possibly any values of the variable in the domain. Equivalent Equations are different equations that have the same solution. Equivalent Equations can be generated by rearranging an equation using the Properties of Equality Aug 7 3:03 PM SUMMARY: How to solve linear equations Apply the properties of equality to... generate a series of equivalent equations from the original equation to... isolate the variable on the left side of the equation. Extraneous Solutions Arise from the algebra Do not satisfy the equation. May result from multiplying or dividing an equation by a variable quantity. Are identified by checking the potential solutions using the Substitution Principle Aug 7 3:03 PM 8

9 P.5 Solving Equations What you should learn How to identify a quadratic equation Understand the Zero Factor Property and the role it plays in solving equations How to solve quadratic equations by factoring Aug 5 5:50 PM Quadratic Equations A quadratic equation in x is... ax 2 + bx + c = 0 where a, b, and c are real numbers and a = 0. NOTE: A linear equation is a 1 st degree polynomial equation in x. A quadratic equation is a 2 nd degree polynomial equation in x. Aug 9 4:56 PM 9

10 The Zero Factor Property Aug 9 4:56 PM Example 4 Solving a Quadratic Equation by Factoring Solve the equation by factoring. Oct 4 12:35 AM 10

11 Example 4 Solving a Quadratic Equation by Factoring Solve the equation by factoring. Oct 4 12:35 AM P.5 Solving Equations What you should learn How to solve quadratic equations using the square root principle Aug 5 5:50 PM 11

12 The Square Root Principle 4x 2 = 12 Oct 4 12:48 AM The Square Root Principle (x + 3) 2 = 16 Oct 4 12:48 AM 12

13 P.5 Solving Equations What you should learn How to solve quadratic equations by completing the square Aug 5 5:50 PM Completing the Square Pattern for completing the square: = = = Oct 6 11:12 AM 13

14 Example Completing the Square Solve the quadratic equation by completing the square. Oct 6 11:13 AM Example Completing the Square Solve the quadratic equation by completing the square. Oct 6 11:13 AM 14

15 Example Completing the Square Solve the quadratic equation by completing the square. Oct 6 11:13 AM SUMMARY: Completing the Square Group the x terms together (associative property of addition) Factor the leading coefficient out this quantity Square half the coefficient of x 1 and add it inside the quantity. Balance the above step by either of the following: a. b. subtracting the "equivalent" amount outside the quantity on the same side of the equation adding the "equivalent" amount to the other side of the equation Oct 6 11:22 AM 15

16 SUMMARY: Completing the Square (cont'd) 5. Write the trinomial inside the quantity as a squared binomial 6. Solve for x using the square root principle Oct 6 11:22 AM P.5 Solving Equations What you should learn How to solve quadratic equations using the quadratic formula Aug 5 5:50 PM 16

17 Use the method of completing the square to solve a generic quadratic polynomial... Oct 6 4:27 PM Example 6 The Quadratic Formula: Two Distinct Solutions Oct 6 11:11 AM 17

18 Example 7 The Quadratic Formula: One Solution Oct 6 11:11 AM SUMMARY: The Quadratic Formula Starting with the form: Then Oct 6 11:22 AM 18

19 P.5 Solving Equations What you should learn How to solve polynomial equations of higher degree Aug 5 5:50 PM Polynomial Equations of Higher Degree Strategies: Zero Factor Property Factoring Methods used to solve quadratic equations can sometimes be extended to polynomials of higher degree. Never cancel out occurrences of the variable when solving equations!! Oct 6 4:14 PM 19

20 Example 8 Solving a Polynomial Equation by Factoring Oct 10 10:30 AM Example 9 Solving a Polynomial Equation by Factoring Oct 10 10:43 AM 20

21 Prob 152: Recognizing Other Quadratic Forms Oct 10 12:35 PM SUMMARY: Solving Polynomial Equations of Higher Degree Zero Factor Property Must be able to factor the polynomial Can only have zero on the right side of the equation. Put all non zero terms on left side of equation. DO NOT cancel out occurrences of the variable!! Oct 6 11:22 AM 21

22 P.5 Solving Equations What you should learn How to solve radical equations Aug 5 5:50 PM Radical Equations Strategies: Isolate the radical Undo the radical Repeat if necessary Solve for x using previous methods. Extraneous solutions often result from... raising each side of an equation to a power multiplying or dividing each side by the variable Checking your solutions is crucial! Oct 6 4:14 PM 22

23 Example 10 Solving Equations with a Rational Exponent Oct 11 11:27 AM Example 11 Solving Equations Involving Radicals Oct 11 11:27 AM 23

24 Example 11 Solving Equations Involving Radicals Oct 11 11:27 AM SUMMARY: Solving Radical Equations Isolate the (one) radical quantity. Undo the radical by raising both sides to the rationalizing power. If necessary, isolate another radical quantity. Repeat above steps until all radicals are removed. Solve for the variable. Check all solution candidates in the original equation. Oct 6 11:22 AM 24

25 P.5 Solving Equations What you should learn How to solve absolute value equations Aug 5 5:50 PM Absolute Value Equations Recall: Absolute Value means... distance from the origin Strategy: Isolate the absolute value. Set the argument of the absolute value equal to... the other side of the equality AND the opposite of the other side of the equality Oct 6 4:14 PM 25

26 Example Solving an Absolute Value Equation Oct 10 10:30 AM Example 12 Solving an Absolute Value Equation Oct 10 10:30 AM 26

27 Additional Examples Solving Equations Oct 10 10:30 AM Additional Examples Solving Equations Oct 10 10:30 AM 27

28 SUMMARY: Solving Absolute Value Equations Isolate the absolute value. Set the argument of the absolute value equal to... the other side of the equality AND the opposite of the other side of the equality Solve for the variable using previous methods. Oct 6 11:22 AM 28

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