Algebra 2 PDF - Solving Radical Equations

7.5 Name Step It Out Learn the Math Do the Math...

7.5 Name Step It Out Learn the Math Do the Math ――― ――― Solve the equation ​​ √9 − 2x ​​ + 3 = x algebraically. EXAMPLE 1 Solve the equation ​​  √5x + 6 ​​ − 2 = x algebraically. ――― ​​  √5x + 6 ​​ − 2 = x Given ――― ​​  √―― ​​  √9 − 2x ​​ + =x Given 5x + 6 ​​ = x + 2 Isolate the radical. 5x + 6 = (​​​ x + 2)​​  ​​ ――― 2 Square both sides. I solate 5x + 6 = ​​x​  2​​ + 4x + 4 Evaluate the square. ​​  √9 − 2x ​​ = x −  the radical. 0 = ​​x​  ​​ − x − 2 2 Combine like terms. ( )​  2 Square 0 = (x − 2)(x + 1) Factor. − x = ​​​ x − both x = 2, x = −1 Zero Product Rule ​​ sides. ――― Check for extraneous solutions. Evaluate ​​  √5​(2)​ + 6 ​​ − 2 ≟ 2 − x = ​​x​  2​​ − x+  the 2=2 ​​  √―――― square. 5(−1) + 6 ​​ − 2 ≟ −1 Combine −1 = −1 0 = ​​x​  2​​ − x+  like terms. ( 0 = ​​ x − )​​​(x − )​ ​ Factor. Zero x= ,x= Product  Rule Check for extraneous solutions. ―――― √ ( )​ ​​ + 3 ≟ © Houghton Mifflin Harcourt Publishing Company ​​  9 − 2​ ≟ ―――― √ ( )​ ​​ + 3 ≟ ​​  9 − 2​ ≟ Algebra 2 111 Journal and Practice Workbook  Learn the Math Do the Math 3 ―― EXAMPLE 2 Solve the equation Solve the equation 6​​ √ x + 2 ​​ − 18 = 0 algebraically. 3 ――― ​​ √3x − 2 ​​ = 4 algebraically. 3 ――― 3 ―― ​​ √3x − 2 ​​ = 4 Given 6​​ √ x + 2 ​​ − 18 = 0 Given 3 ――― 3 ​​​(​ √3x − 2 ​)​​  ​​ = ​​4​  3​​ Cube both sides. 3 ―― 6​​ √ x + 2 ​​ = Isolate the radical term. 3x − 2 = 64 Evaluate the cube. 3 ―― 3x = 66 Isolate the variable term. ​​ √ x + 2 ​​ = Divide. x = 22 Divide. 3 ―― 3 ​​​(​ √ x + 2 ​)​​  ​​= ​​ 3 ​ ​​ Cube both sides. Check the solution. 3 ―――― ​​ √ 3(22) − 2 ​​ ≟ 4 x+ 2= Evaluate the cube. 4=4 x= Subtract. Check the solution. ―――― 6​​  √ 3 + 2 ​​ − 18 ≟ 0 ≟0 Learn the Math EXAMPLE 3 ― The function t = ​​  √__ d ​  16 ​ ​​relates the approximate time, in seconds, it takes for an object to fall f feet. An acorn falls for 1.5 seconds before it hits the ground. How far does the acorn fall? ―― √​  d  ​ ​​ 1.5 = ​​  _ 16 Replace t with 1.5. ―― 2.25 = (​​​ ​  √_ ​  d  ​ ​)​​  ​​ © Houghton Mifflin Harcourt Publishing Company 2 Square both sides. 16 ​​  d  ​​ 2.25 = _ Simplify. 16 36 = d Multiply both sides by 16. The acorn falls 36 feet in 1.5 seconds. Do the Math ―― The function v = ​​  √21d ​​ relates the speed, in miles per hour, of a bicycle before applying its brakes if the skid marks are d feet long. Find the length of a skid for a bicycle at 19 miles per hour. Round your answer to the nearest tenth. Algebra 2 112 Journal and Practice Workbook  LESSON 7.5 Name More Practice ONLINE ONL IN E Video Tutorials and Video Tutorials and Interactive Examples Solve each equation. Show each step of your solution. Interactive Examples Check for extraneous roots. 1. ――― ​​  √2x − 1 ​​ + 2 = x 2. ― ​​  √3x ​​ + 10 = x + 4 3. ――― ​​  √20 − 8x ​​ = x 4. ―― ​​  √x + 3 ​​ = x − 3 5. ――― x − 1 = ​​  √x + 11 ​​ 6. ​  1 ​ _ ​​​(3x − 5)​​  2 ​​ = 10 3 ―― 3 ――― 7. ​​ √x − 7 ​​ + 2 = 5 8. ​​ √​x​  2​ − 19 ​​ = 5 © Houghton Mifflin Harcourt Publishing Company 3 ――― ​  1 ​ _ 9. ​​ √ 5x + 3 ​​ − 9 = − 2 10. ​​​(9 − 3x)​​  3 ​​ = − 6 Algebra 2 113 Journal and Practice Workbook  ――― 11. Explain how you know that ​​ √4 − 3x ​​ + 5 = 2 has no solutions in the real number system without solving the equation. ―― 12. Math on the Spot The speed s in miles per hour that a car is traveling when it goes into a skid can be estimated by using the formula s = ​​  √30fd ​​ where f is the coefficient of friction and d is the length of the skid marks in feet. After an accident, a driver claims to have been traveling the speed limit of 55 miles per hour. The coefficient of friction under accident conditions was 0.8. Is the driver telling the truth about his speed if the length of the actual skid marks is 125 feet? Explain. ― 13. The function t = 2π​​  √__ d ​  32 ​ ​​ models the time t in seconds for a pendulum d feet long to complete one swing back and forth. How long is a pendulum that completes one swing in 1.5 seconds? Round to the nearest hundredth. Show your work. 3 ― 14. The function l = 46​​ √ m ​​ models the length l in centimeters of a certain type of © Houghton Mifflin Harcourt Publishing Company fish with a mass of m kilograms. What is the mass of a fish of this type that is 50 centimeters long? Round to the nearest hundredth. Show your work. 15. The diameter of a field__1 hockey ball is about 7.3 centimeters. The radius of a sphere ( 4π ) with volume V is ​​​ ___ ​  ​ ​  3V ​ ​​  3 ​​. What is the volume in cubic centimeters of a field hockey ball? Round to the nearest tenth. A 203.7 B 407.4 C 814.8 D 1629.5 Algebra 2 114 Journal and Practice Workbook