Precalc0011to0016

# Precalc0011to0016 - (Section 0.11 Solving Equations 0.11.1...

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(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES • Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION • Much of precalculus is devoted to solving equations of various types. In this section, we will focus on solving basic algebraic equations. • We will solve polynomial equations more generally in Chapter 2, exponential and logarithmic equations in Chapter 3, and trigonometric and inverse trigonometric equations in Chapters 4 and 5. We will solve systems of equations in Chapters 7 and 8. PART B: SOLVING EQUATIONS A solution to an equation in x is a number that makes the equation true when the number is substituted for x . • For now, we only consider real solutions. We solve an equation by finding its solution set , the set of all solutions. When solving an equation, we often write a sequence of equivalent equations , which have the same solution set . • Adding, subtracting, multiplying, and dividing the same nonzero number on both sides of an equation maintains equivalence.

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(Section 0.11: Solving Equations) 0.11.2 Example Set 1 (Solving Equations) When we solve the linear equation 2 x = 6 , we divide both sides by 2 and obtain the equivalent equation x = 3 . The solution set of both equations is 3 {} . We can check a solution by verifying that it satisfies the original equation : 23 () = 6 , so 3 checks out. The equation x = x + 1 has no real solutions. Its solution set is ± , the empty set (or null set ). The equation x + 1 = x + 1 is solved by all real numbers. Its solution set is ± , and the equation is called an identity . § WARNING 1 : There is a difference between simplifying an expression and solving an equation . For example, when we simplify the expression 2 x + x , we write “ 2 x + x = 3 x ,” and 3 x is our answer. On the other hand, when we solve the equation 2 x + x = 3 x , we state that the solution set is ± . PART C: SOLVING QUADRATIC EQUATIONS The general form of a quadratic equation in x is given by: ax 2 + bx + c = 0 , where a ± 0 , and a , b , and c are real coefficients. Its solutions are given by the Quadratic Formula : x = ± b ± b 2 ± 4 ac 2 a • Sometimes, the solutions are not real, but imaginary (see Section 2.1.) WARNING 2 : Make sure the fraction bar in the formula goes all the way across. The formula is not : x = ± b ± b 2 ± 4 ac 2 a . WARNING 3 : Here, the plus-minus sign ± indicates that we take both the result from the “+” case and the result from the “ ± ” case. (The results are equal ± b 2 ± 4 ac = 0 .) Sometimes in precalculus, the ± sign indicates that we do not yet know which sign to take. We also use the Factoring Method, the Square Root Method, and the Completing the Square (CTS) Method to solve quadratic equations.
(Section 0.11: Solving Equations) 0.11.3 Example 2 (Using the Quadratic Formula) Solve the equation 2 x 2 ± 7 x = 15 using the Quadratic Formula.

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Precalc0011to0016 - (Section 0.11 Solving Equations 0.11.1...

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