6.6 Solve Radical Equations Goal p Solve radical equations. Your Notes VOCABULARY Radical equation SOLVING RADICAL EQUATIONS To solve a radical equation, follow these steps: the radical on one side of the equation, if necessary. Step 1 Step 2 Raise each side of the equation to the same to eliminate the radical and obtain a linear, quadratic, or other polynomial equation. the polynomial equation using techniques you learned in previous chapters. Check your solution. Step 3 Example 1 Solve a radical equation } Solve Ïx 1 6 5 3. } Ïx 1 6 5 3 Write original equation. 5 Square each side to eliminate the radical. 5 Simplify. 5 Subtract The solution is from each side. . Check this in the original equation. Checkpoint Solve the equation. Check your solution. 3} 1. Ïx 2 5 1 1 5 21 178 Lesson 6.6 • Algebra 2 Notetaking Guide Copyright © Holt McDougal. All rights reserved. 6.6 Solve Radical Equations Goal p Solve radical equations. Your Notes VOCABULARY Radical equation An equation with a radical that has variables in the radicand SOLVING RADICAL EQUATIONS To solve a radical equation, follow these steps: Step 1 Isolate the radical on one side of the equation, if necessary. Step 2 Raise each side of the equation to the same power to eliminate the radical and obtain a linear, quadratic, or other polynomial equation. Step 3 Solve the polynomial equation using techniques you learned in previous chapters. Check your solution. Example 1 Solve a radical equation } Solve Ïx 1 6 5 3. } Ïx 1 6 5 3 } ( Ï x 1 6 ) 2 5 32 x16 5 9 x 5 3 Write original equation. Square each side to eliminate the radical. Simplify. Subtract 6 from each side. The solution is 3 . Check this in the original equation. Checkpoint Solve the equation. Check your solution. 3} 1. Ïx 2 5 1 1 5 21 23 178 Lesson 6.6 • Algebra 2 Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Solve an equation with a rational exponent Example 2 (3x 1 4)2/3 5 16 Original equation Raise each side to the 5 3 power }2 . 5 Apply properties of exponents. 5 Simplify. 5 Subtract 5 Divide each side by . . Check this in the original equation. The solution is Solve an equation with an extraneous solution Example 3 } x 2 2 5 Ï x 1 10 50 CHECK Check x 5 Original equation 5 Square each side. 5 Expand left side and simplify right side. 50 Write in standard form. 50 Factor. or x5 or . } x 2 2 5 Ï x 1 10 0 0 The only solution is extraneous.) Copyright © Holt McDougal. All rights reserved. from each side. 50 x5 Zero product property Solve for x. Check x 5 2 . } x 2 2 5 Ï x 1 10 0 0 . (The apparent solution Lesson 6.6 • Algebra 2 Notetaking Guide is 179 Your Notes Solve an equation with a rational exponent Example 2 (3x 1 4)2/3 5 16 Original equation Raise each side to the [(3x 1 4)2/3]3/2 5 163/2 3 power }2 . 3x 1 4 5 (161/2)3 Apply properties of exponents. 3x 1 4 5 64 Simplify. 3x 5 60 Subtract 4 from each side. x 5 20 Divide each side by 3 . The solution is 20 . Check this in the original equation. Solve an equation with an extraneous solution Example 3 } x 2 2 5 Ï x 1 10 Original equation } (x 2 2)2 5 (Ï x 1 10 )2 Square each side. x 2 2 4x 1 4 5 x 1 10 Expand left side and simplify right side. x 2 2 5x 2 6 5 0 Write in standard form. (x 2 6)(x 1 1) 5 0 Factor. x26 50 Zero product property or x 1 1 5 0 x 5 6 or CHECK Check x 5 6 . } x 2 2 5 Ï x 1 10 } 6 2 2 0 Ï6 1 10 } 4 0 Ï16 4 5 4 x 5 21 Solve for x. Check x 5 2 1 . } x 2 2 5 Ï x 1 10 } 21 2 2 0 Ï 21 1 10 } 23 0 Ï 9 23 Þ 3 The only solution is 4 . (The apparent solution 21 is extraneous.) Copyright © Holt McDougal. All rights reserved. Lesson 6.6 • Algebra 2 Notetaking Guide 179 Your Notes Example 4 Solve an equation with two radicals } } Solve Ïx 1 6 1 2 5 Ï 10 2 3x . } } Ïx 1 6 1 2 5 Ï10 2 3x 5 5 5 50 x5 CHECK 5 Divide each side by 4. 5 Square each side again. 5 Simplify. 05 Write in standard form. 05 Factor. 50 or x5 or Check x 5 Write original equation. Square each side. Expand left side and simplify right side. Isolate radical expression. Zero product property Solve for x. . 0 0 5 Check x 5 2 . 0 0 5 The only solution is is extraneous.) 180 Lesson 6.6 • Algebra 2 Notetaking Guide . (The apparent solution Copyright © Holt McDougal. All rights reserved. Your Notes Solve an equation with two radicals Example 4 } } Solve Ïx 1 6 1 2 5 Ï 10 2 3x . } } Ïx 1 6 1 2 5 Ï10 2 3x } } (Ïx 1 6 1 2)2 5 (Ï10 2 3x )2 } x 1 6 1 4Ïx 1 6 1 4 5 10 2 3x } 4Ï x 1 6 5 24x } Ïx 1 6 5 2x } (Ïx 1 6 )2 5 (2x)2 x 1 6 5 x2 0 5 x2 2 x 2 6 Write original equation. Square each side. Expand left side and simplify right side. Isolate radical expression. Divide each side by 4. Square each side again. Simplify. Write in standard form. 0 5 (x 2 3)(x 1 2) Factor. x23 50 or x 5 3 or CHECK x12 50 x 5 22 Zero product property Solve for x. Check x 5 3 . } } Ï3 1 6 1 2 0 Ï10 2 3(3) } } Ï9 1 2 0 Ï1 5 5 1 Check x 5 2 22 . } }} Ï(22)1 6 1 2 0 Ï10 2 3(22) } } Ï4 1 2 0 Ï16 4 5 4 The only solution is 22 . (The apparent solution 3 is extraneous.) 180 Lesson 6.6 • Algebra 2 Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Checkpoint Solve the equation. Check for extraneous solutions. 2. 22x 4/3 2 21 5 253 } 3. x 1 2 5 Ï2x 1 7 } } 4. Ï 3x 1 4 2 1 5 Ï x 1 5 Homework Copyright © Holt McDougal. All rights reserved. Lesson 6.6 • Algebra 2 Notetaking Guide 181 Your Notes Checkpoint Solve the equation. Check for extraneous solutions. 2. 22x 4/3 2 21 5 253 8 } 3. x 1 2 5 Ï2x 1 7 1 } } 4. Ï 3x 1 4 2 1 5 Ï x 1 5 4 Homework Copyright © Holt McDougal. All rights reserved. Lesson 6.6 • Algebra 2 Notetaking Guide 181

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