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5-59. BACKYARD SOLUTIONS a. b. Draw a diagram and write an equation to represent each scenario below. Solve the equation, and then explain whether the solutions make sense. Ernie is going to install a square (x + 3)2 = 169 hot tub in his backyard. He plans |x + 3| = 13 to add a 3-foot-wide deck on x + 3 = 13 or x + 3 = -13 two adjacent sides of the hot x = 10 or x = -16 tub. If Ernie’s backyard (which is 10 feet x 10 feet hot tub also a square) has 169 square feet of space, what is the largest hot tub he can order? (x + 1)2 = 12 Gabi is creating a decorative rock |x + 1| = 12 garden that will cover 12 square Rock .5 ft x + 1 =2 3 or x + 1 = -2 3 feet of her back yard, including its .5 ft garden frame. She plans to build a 6-inch x = -1+ 2 3 or x = -1- 2 3 wide wood frame around the x = 2.5 or x = -4.5 Wood frame rock garden to keep the rocks in 2.5 feet x 2.5 feet rock garden place. If the framed rock garden is square, how much area will Gabi need to cover with decorative rocks? 5.2.1 Perfect Square Equations January 25, 2016 Objectives CO: SWBAT solve quadratic equations in perfect square form and express their solutions in exact and approximate forms. LO: SWBAT explain how many solutions a quadratic equation written in perfect square form has. 5-60. In problem 5-59, did you notice anything special about the forms of the equations you wrote? Did you use the Zero Product Property to solve your equations, or were you able to solve them a different way? Without rewriting, determine the value(s) of x that make(s) each equation true. a. (x – 1)2 = 4 b. (x – 1)2 = 0 c. (x – 1)2 = –4 |x – 1| = 2 |x – 1| = 0 x – 1 = 2 or x – 1 = -2 x – 1 = 0 x=3 or x = -1 x=1 d. What method did you use No solution because no number times itself will ever give a negative. to solve each equation? 5-61. The quadratic equation (x – 3)2 = 12 is written in perfect square form. It is called this because the expression (x – 3)2 forms a square when built with algebra tiles. a. Solve (x – 3)2 = 12. Write your answer in exact form (or radical form). That is, write it in a form that is precise and does not have any rounded decimals. (x – 3)2 = 12 |x – 3| = 12 x – 3 = 2 3 or x – 3 = - 2 3 x = 3 + 2 3 or x =3 - 2 3 How many solutions are there? b. 2 c. The solution(s) from part (a) are irrational.That is, they are decimals that never repeat and never end. Write the solution(s) in approximate decimal form. Round your answers to the nearest hundredth. x ≈ 6.46 or –0.46 5-62. THE NUMBER OF SOLUTIONS The equation in problem 5-61 had two solutions. However, in problem 5-60, you saw that a quadratic equation might have one solution or no solutions at all. How can you determine the number of solutions to a quadratic equation? With your team, solve the equations below. Express your answers in both exact form and approximate form. Look for patterns among the equations with no solution and those with only one solution. Be ready to report your patterns to the class. a. (2x – 3)2 = 49 x = –2 or 5 2 solutions b. (7x – 5)2 = –2 No solution c. d. (5 – 10x)2 = 0 x=½ 1 solution e. (x + 2)2 = –10 No solution f. 2 solutions → squared equals positive 1 solution → squared equals zero No solution → squared equals negative (x + 4)2 = 20 x = 3 + 2 3 or 3 - 2 3 x ≈ 0.47 or –8.37 2 solutions (x + 11)2 + 5 = 5 x = -11 1 solution