1. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Solving Radical Equations and Inequalities Holt Algebra 2 Skills Check Skills Check Lesson Presentation Lesson Presentation Holt McDougal Algebra 2

2. Solving Radical Equations and Inequalities Solve radical equations. Objective

3. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities A radical equation contains a variable within a radical. Recall that you can solve quadratic equations by taking the square root of both sides. Similarly, radical equations can be solved by raising both sides to a power.

4. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities For a square root, the index of the radical is 2. Remember!

5. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Solve each equation. Example 1: Solving Equations Containing One Radical Subtract 5. Simplify. Square both sides. Solve for x. Simplify. Check

6. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities 7 Example 2: Solving Equations Containing One Radical Divide by 7. Simplify. Cube both sides. Solve for x. Simplify. Check Solve each equation. 3 7 7 5x  7 84 7 

7. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Example 3: Solving Equations Containing Two Radicals Square both sides. Solve Simplify. Distribute. Solve for x. 7x + 2 = 9(3x – 2) 7x + 2 = 27x – 18 20 = 20x 1 = x

8. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Cube both sides. Solve each equation. Simplify. Distribute. x + 6 = 8(x – 1) You Try! Example 4 x + 6 = 8x – 8 14 = 7x 2 = x Check 2 Solve for x.

9. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Raising each side of an equation to an even power may introduce extraneous solutions. You don’t have to worry about extraneous solutions when solving problems to an odd power.

10. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Example 5 Step 1 Solve for x. Square both sides. Solve for x. Factor. Write in standard form. Simplify. –3x + 33 = 25 – 10x + x 2 0 = x 2 – 7x – 8 0 = (x – 8)(x + 1) x – 8 = 0 or x + 1 = 0 x = 8 or x = –1

11. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Example 5 Continued Method 2 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. 3 –36 x Because x = 8 is extraneous, the only solution is x = – 1.

12. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Step 1 Solve for x. Square both sides. Solve for x. Factor. Write in standard form. Simplify. 2x + 14 = x 2 + 6x + 9 0 = x 2 + 4x – 5 0 = (x + 5)(x – 1) x + 5 = 0 or x – 1 = 0 x = –5 or x = 1 You Try! Example 6

13. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. 4 x Because x = – 5 is extraneous, the only solution is x = 1. You Try! Example 6 Continued 2 –2

14. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Solve for x. Factor. Write in standard form. Simplify. –9x + 28 = x 2 – 8x + 16 0 = x 2 + x – 12 0 = (x + 4)(x – 3) x + 4 = 0 or x – 3 = 0 x = –4 or x = 3 Example 7

15. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. Example 7 Continued So BOTH answers work!!! x = –4 or x = 3

16. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Example 8: Solving Equations with Rational Exponents Solve each equation. (5x + 7) = 3 Cube both sides. Solve for x. Factor. Simplify. 5x + 7 = 27 5x = 20 x = 4 1 3

17. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Example 9: Solving Equations with Rational Exponents Factor. Solve for x. Factor out the GCF, 4. Raise both sides to the reciprocal power. Simplify. 2x = (4x + 8) 1 2 (2x) 2 = [(4x + 8) ] 2 1 2 4x 2 = 4x + 8 Write in standard form. 4x 2 – 4x – 8 = 0 4(x 2 – x – 2) = 0 4(x – 2)(x + 1 ) = 0 4 ≠ 0, x – 2 = 0 or x + 1 = 0 x = 2 or x = – 1 Step 1 Solve for x.

18. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Example 9 Continued Step 2 Use substitution to check for extraneous solutions. 2x = (4x + 8) 1 2 2(2) (4(2) + 8) 1 2 4 16 1 2 4 2x = (4x + 8) 1 2 2(–1) (4(–1) + 8) 1 2 –2 4 1 2 –2 2 x The only solution is x = 2.

19. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Solve for x. Raise both sides to the reciprocal power. Simplify. 9(x + 6) = 81 [3(x + 6) ] 2 = (9) 2 1 2 Distribute 9. 9x + 54 = 81 Example 10 9x = 27 x = 3 3(x + 6) = 9 1 2 Simplify.

20. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Cwk/Hwk Pg 240 2-22 2 nd 2 columns, 54

21. Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Lesson Quiz: Part I Solve each equation or inequality. x = 16 1. 2. 3. 4. x = –5 x = 14 x = 2 x ≥ 59 5.