Quadratics and inequalities

3 add to each side the square of 1 2 the coefficient

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3. Add to each side the square of 1 2 the coefficient of x . 4. Factor the left-hand side as the square of a binomial. 5. Apply the even-root property. 6. Solve for x . 7. Simplify. Study Tip Most instructors believe that what they do in class is important. If you miss class, then you miss what is important to your instructor and what is most likely to appear on the test. Calculator Close-Up Note that the x -intercepts for the graph of the function y 2 x 2 3 x 2 are ( 2, 0) and 1 2 , 0 : 6 4 2 6 Completing the square with a 1 Solve 2 x 2 3 x 2 0 by completing the square. Solution For completing the square, the coefficient of x 2 must be 1. So we first divide each side of the equation by 2: 2 x 2 2 3 x 2 0 2 Divide each side by 2. x 2 3 2 x 1 0 Simplify. x 2 3 2 x 1 Get only x 2 and x terms on the left-hand side. x 2 3 2 x 1 9 6 1 1 9 6 One-half of 3 2 is 3 4 , and 3 4 2 1 9 6 . x 3 4 2 2 1 5 6 Factor the left-hand side. x 3 4 2 1 5 6 Even-root property x 3 4 5 4 or x 3 4 5 4 x 2 4 1 2 or x 8 4 2 Check these values in the original equation. The solution set is 2, 1 2 . Now do Exercises 49–50 dug22241_ch10a.qxd 11/10/2004 18:30 Page 622
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In Examples 5 and 6 the solutions were rational numbers, and the equations could have been solved by factoring. In Example 7 the solutions are irrational numbers, and factoring will not work. Math at Work Financial Matters In the United States over 1 million new homes are sold annually, with a median price of about $200,000. Over 17 million new cars are sold each year with a median price over $20,000. Americans are constantly saving and borrowing. Nearly everyone will need to know a monthly payment or what their savings will total over time. The answers to these questions are in the following table. What $ P Left at Compound What $ R Deposited Periodic Payment That Will Interest Will Grow to Periodically Will Grow to Pay off a Loan of $ P P (1 i ) nt R (1 i ) i nt 1 P 1 (1 i i ) nt Monthly payment ($) 2000 1000 1500 500 2 4 10 8 6 APR (percent) 20-year $200,000 mortgage 0 In each case n is the number of periods per year, r is the annual percentage rate (APR), t is the number of years, and i is the interest rate per period i n r . For periodic payments or deposits these expressions apply only if the compounding period equals the payment period. So let’s see what these expressions do. A person inherits $10,000 and lets it grow at 4% APR compounded daily for 20 years. Use the first expression with n 365, i 0 3 . 6 0 5 4 , and t 20 to get 10,000 1 0 3 .0 6 4 5 365 20 or $22,254.43, which is the amount after 20 years. More often, people save money with periodic deposits. Suppose you deposit $100 per month at 4% compounded monthly for 20 years. Use the second expression with R 100, i 0 1 .0 2 4 , n 12, and t 20 to get 100 or $36,677.46, which is the amount after 20 years. Suppose that you get a 20-year $200,000 mortgage at 7% APR compounded monthly to buy an average house. Try using the third expression to calculate the monthly payment of $1550.60. See the accompanying figure.
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