Solving Radical Equations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Radical Equations Equations containing variables within radical signs are called radical equations. A solution to a radical equation is a real number which, when substituted for the variable, gives a true equation. Examples:1. Show that 12 is a solution of.2. Show that 0 is a solution of.3. Show that has no solutions. (12)  True. (0)  True. Since must be positive, has no solutions. The symbol indicates the positive or principal square root of a number.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Property of Raising Each Side of an Equation to a Power If a and b are real numbers, n is a positive integer, and a = b, then a n = b n This principle can be applied to solve radical equations. Examples: 1. If, then  x = If, then  x = 125. The converse of the statement is true for odd n and false for even n. If n is odd and a n = b n then a = b. If n is even and a n = b n then it is possible that a  b. Example: (  3) 2 = 9 = (3) 2 but,  3  3.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Radical Equations With One Square Root To solve a radical equation containing one square root: Example: Solve. Original equation Isolate the square root. Square both sides. Solve for x. Check. (22) True 4. Check the solutions. 3. Solve for the variable. 2. Square both sides of the equation. 1. Isolate the radical on one side of the equation.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Solve Example: Solve. Original equation Isolate the square root. Square both sides. Solve for x. Solutions Check.True (1) (9) Check.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 (x 2 + 3) does not give a root because the square root of a negative number is not a real number. Example: Solve. Example: Solve Original equation Isolate the cube root. Cube both sides. Expand the cube. Solution Simplify. Group terms. Factor. Check. (9) True

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Radical Equations With Two Square Roots To solve a radical equation containing two square roots: 5. Check the solutions. 4. Solve the equation. 3. Isolate the other radical. 2. Square both sides of the equation. 1. Isolate one radical on one side of the equation. Example: Solve. Isolate one radical. Square both sides. Isolate the other radical. Simplify. Square both sides. Check. (1) Solution True

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Solve Example: Solve. Original equation Isolate the square root. Square both sides. Simplify. Factor. Possible solutions Check 2. (2) 2 is a solution of the original radical equation. True Check 14. (14) 14 is not a solution of the original radical equation. False

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Both are solutions of the radical equation, but since the distance from the bottom of the board to the wall must be nonnegative, – 6 is not a solution of the problem. Example: Word Problem Example: A ten-foot board leans against an 8-foot wall so that the top end of the board is at the top of the wall. How far must the bottom of the board be from the wall? Let x be the distance from the bottom of the board to the wall. Check. The bottom of the board must be 6 feet from the wall. Simplify. Use the Pythagorean Theorem. Possible solutions 8 ft.wall x 10 ft. board

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Word Problem Example: The time T (in seconds) taken for a pendulum of length L (in feet) to make one full swing, back and forth, is given by the formula To the nearest hundredth, how long is a pendulum which takes 2 seconds to complete one full swing? A pendulum of length approximately 3.24 feet will make one full swing in 2 seconds. Radical equation Isolate the radical. Square both sides. Solve for L. (to the nearest hundredth) Check. (3.24) True.

Teach Us Math Mr. Whitt… mub9ohttp:// mub9o Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

Cross Number Puzzle Solve each equation Be sure to fill out the correct answer in the corresponding space Check for extraneous solutions When finished, try the proof on the back page! Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

Exit Ticket Complete the following Algebraic Proof You will see these A LOT in Geometry!!! Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

Homework “Solving Radical Equations” Handout Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14