How can you determine the number of solutions of a quadratic equation of the form ax 2 + c = 0? ACTIVITY: The Number of Solutions of ax 2 + c = 0
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1 9. Solving Quadratic Equations Using Square Roots How can ou determine the number of solutions of a quadratic equation of the form a + c = 0? ACTIVITY: The Number of Solutions of a + c = 0 Work with a partner. Solve each equation b graphing. Eplain how the number of solutions of a + c = 0 relates to the graph of = a + c. Quadratic equation Quadratic function a. = 0 b. + = c. = 0 d. = 0 CMMN CRE Solving Quadratic Equations In this lesson, ou will solve quadratic equations b taking square roots. Learning Standard A.REI.b Chapter 9 Solving Quadratic Equations
2 ACTIVITY: Estimating Solutions Work with a partner. Complete each table. Use the completed tables to estimate the solutions of = 0. Eplain our reasoning. a. b ACTIVITY: Using Technolog to Estimate Solutions Math Practice Choose Appropriate Tools What different tpes of technolog can be used to answer the questions? Which tool would be the most appropriate and wh? Work with a partner. Two equations are equivalent when the have the same solutions. a. Are the equations = 0 and = equivalent? Eplain our reasoning. b. Use the square root ke on a calculator to estimate the solutions of = 0. Describe the accurac of our estimates. c. Write the eact solutions of = 0.. IN YUR WN WRDS How can ou determine the number of solutions of a quadratic equation of the form a + c = 0?. Write the eact solutions of each equation. Then use a calculator to estimate the solutions. a. = 0 b. = 0 c. = 8 Use what ou learned about quadratic equations to complete Eercises on page. Section 9. Solving Quadratic Equations Using Square Roots
3 9. Lesson Lesson Tutorials In Section., ou studied properties of square roots. Here ou will use square roots to solve quadratic equations of the form a + c = 0. Solving Quadratic Equations Using Square Roots You can solve = d b taking the square root of each side. When d > 0, = d has two real solutions, = ± d. When d = 0, = d has one real solution, = 0. When d < 0, = d has no real solutions. EXAMPLE Solving Quadratic Equations Using Square Roots a. Solve 7 = 0 using square roots. 7 = 0 Write the equation. = 7 Add 7 to each side. = 9 Divide each side b. = ± 9 = ± Take the square root of each side. Simplif. The solutions are = and =. b. Solve 0 = 0 using square roots. 0 = 0 Write the equation. = 0 Add 0 to each side. = 0 Take the square root of each side. The onl solution is = 0. Remember The square of a real number cannot be negative. That is wh the equation in part (c) has no real solutions. c. Solve + = using square roots. + = Write the equation. = Subtract from each side. = Divide each side b. The equation has no real solutions. Eercises 0 Solve the equation using square roots.. = 7. + = 0. = Chapter 9 Solving Quadratic Equations
4 EXAMPLE Solving a Quadratic Equation Using Square Roots Solve ( ) = using square roots. ( ) = Write the equation. = ± Take the square root of each side. = ± Add to each side. So, the solutions are = + = and = =. Check 0 Use a graphing calculator to check our answer. Rewrite the equation as ( ) = 0. Graph the related function = ( ) and find the -intercepts, or zeros. The zeros are and, so the solution checks EXAMPLE Real-Life Application A touch tank has a height of feet. Its length is times its width. The volume of the tank is 70 cubic feet. Find the length and width of the tank. feet The length is times the width w, so = w. Write an equation using the formula for the volume of a rectangular prism. V = wh Write the formula. Stud Tip Use the positive square root because negative solutions do not make sense in this contet. Length and width cannot be negative. 70 = w(w)() Substitute 70 for V, w for, and for h. 70 = 9w Multipl. 0 = w Divide each side b 9. ± 0 = w Take the square root of each side. The solutions are 0 and 0. Use the positive solution. So, the width is 0. feet and the length is 0. feet. Eercises 8 and Solve the equation using square roots.. ( + 7) = 0. ( ) = 9. ( + ) = 7. WHAT IF? In Eample, the volume of the tank is cubic feet. Find the length and width of the tank. Section 9. Solving Quadratic Equations Using Square Roots
5 9. Eercises Help with Homework. REASNING How man real solutions does the equation = d have when d is positive? 0? negative?. WHICH NE DESN T BELNG? Which equation does not belong with the other three? Eplain our reasoning. = 9 = = 7 = 9+(-)= +(-)= +(-9)= 9+(-)= Determine the number of solutions of the equation. Then use a calculator to estimate the solutions.. = = 0. = 0 Determine the number of solutions of the equation. Then solve the equation using square roots.. = 7. = 8. = 8 9. = 0. = 0. = 9 Solve the equation using square roots.. = 0. + = 0. + = 0. = = = = 7 9. = = 8. ERRR ANALYSIS Describe and correct the error in solving the equation. Solve = 9. = 7 Add to each side. = Divide each side b. = The solution is =. Take the square root of each side.. WAREHUSE A bo falls off a warehouse shelf from a height of feet. The function h = + gives the height h (in feet) et) of the bo after seconds. When does it hit the floor? Chapter 9 Solving Quadratic Equations
6 Solve the equation using square roots. Use a graphing calculator to check our solution(s).. ( + ) = 0. ( ) =. ( ) = 8. ( ) = ( + ) = 8. ( ) = Use the given area A to find the dimensions of the figure. 9. A = in. 0. A = 78 cm. A = π ft. PND An in-ground pond has the shape of a rectangular prism. The pond has a height of inches and a volume of,000 cubic inches. The pond s length is times its width. Find the length and width of the pond. ft. AREA RUG The design of a square area rug for our living room is shown. You want the area of the inner square to be % of the total area of the rug. Find the side length of the inner square. ft. WRITING How can ou approimate the roots of a quadratic equation when the roots are not integers?. LGIC Given the equation a + c = 0, describe the values of a and c so the equation has the following number of solutions. a. two solutions b. one solution c. no solutions. Without graphing, where do the graphs of = and = 9 intersect? Eplain. Find the product. (Section 7.) 7. ( + ) 8. (w 7) 9. ( ) 0. MULTIPLE CHICE What is an eplicit equation for a =, a n = a n +? (Section.7) A a n = n B a n = n C a n = n + D a n = n + Section 9. Solving Quadratic Equations Using Square Roots 7
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