Digital Lesson Solving Radical Equations Equations containing variables within radical signs are called radical equations. 3 x 3 3 x 9 x 7 x 1 2 3 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 x 3 3 . (12) 3 3 9 3 True. 2. Show that 0 is a solution of 3 x 1 2 3 . 3 (0) 1 2 3 3 1 2 3 1 2 3 True. 3. Show that x 3 has no solutions. The symbol indicates the positive or principal square root of a number. Since x must be positive, x 3 has no solutions. Copyright by Houghton Mifflin Company, Inc. All rights reserved.
2 If a and b are real numbers, n is a positive integer, and a = b, then an = bn This principle can be applied to solve radical equations. x 2 , then ( x ) 2 2 2 x = 4. Examples: 1. If 2. If 3 x 5, then (3 x )3 53 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. 3
To solve a radical equation containing one square root: 1. Isolate the radical on one side of the equation. 2. Square both sides of the equation. 3. Solve for the variable. 4. Check the solutions. Example: Solve x 3 5 0. x 3 5 0 x 3 5 x 3 25 x 22 Original equation Isolate the square root. Square both sides. Solve for x. (22) 3 5 25 5 5 5 0 Check. True Copyright by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Solve 4 x x 3. 4 x x 3
4 x x 3 Original equation Isolate the square root. 16 x ( x 3) 2 Square both sides. 16 x x 2 6 x 9 Solve for x. x 2 10 x 9 0 ( x 1)( x 9) 0 x 1 or x 9 4 (1) (1) 4 1 3 Solutions Check. True 4 (9) (9) 4(3) 9 12 9 3 Check. True Copyright by Houghton Mifflin Company, Inc. All rights reserved. 5
3 Example: Solve 2 3 x 3 x. 2 3 3x 3 x 2 3 3x x 3 8(3 x) ( x 3)3 Original equation Isolate the cube root. Cube both sides. 24 x x 3 9 x 2 27 x 27 Expand the cube. x 3 9 x 2 3x 27 0 x 2 ( x 9) 3( x 9) 0 ( x 2 3)( x 9) 0 (x2 + 3) does not give a root because the square root of a negative number is not a real number. x 9 2 3 3 (9) 3 2 3 27 3 2(3) 3 9 Copyright by Houghton Mifflin Company, Inc. All rights reserved.
Simplify. Group terms. Factor. Solution Check. True 6 To solve a radical equation containing two square roots: 1. Isolate one radical on one side of the equation. 2. Square both sides of the equation. 3. Isolate the other radical. 4. Solve the equation. 5. Check the solutions. Example: Solve x 8 x 8 4 x 8 (4 x 8 16 8 x 8 64 x 64 x 1 (1) 8 (1) 4 Copyright by Houghton Mifflin Company, Inc. All rights reserved. x 4 .
x Isolate one radical. x )2 Square both sides. 8 x x Simplify. Isolate the other radical. Square both sides. Solution Check. True 7 Example: Solve x 2 x 2 6 . x 2 x 2 6 Original equation 2 x 2 6 x Isolate the square root. 4( x 2) 36 12 x x 2 Square both sides. x 2 16 x 28 0 ( x 2)( x 14) 0 x 2 or x 14 Simplify. Factor.
Possible solutions (2) 2 (2) 2 2 2 4 2 2(2) 6 Check 2. True 2 is a solution of the original radical equation. (14) 2 (14) 2 14 2 16 14 8 22 6 Check 14. False 14 is not a solution of the original radical equation. Copyright by Houghton Mifflin Company, Inc. All rights reserved. 8 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. Use the Pythagorean Theorem. 10 ft. board x 2 82 10 2 x 2 100 64 36 Simplify. x 6 or x 6 Possible solutions 8 ft.wall x
Check. 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. The bottom of the board must be 6 feet from the wall. Copyright by Houghton Mifflin Company, Inc. All rights reserved. 9 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 L T 2 . 32 To the nearest hundredth, how long is a pendulum which takes 2 seconds to complete one full swing? L 2 2 2 (3.24) 1.99 Check. True. Radical equation 32 32 L 2
Isolate the radical. A pendulum of length 32 2 approximately 3.24 feet will 2 L 1 make one full swing in 2 Square both sides. 32 seconds. 32 L 2 3.24 Solve for L. (to the nearest hundredth) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 10
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