1. Tom Worthing Intermediate Algebra All rights reserved. 1 Higher order equations Use Factoring method or calculator Know how many roots you will have Determine the type of roots Real roots, rational or irrational Complex non real roots

2. Tom Worthing Intermediate Algebra All rights reserved. 2 Ex. 1 Solve x 4 – 36x 2 = 0 (4 possible) x = 0, x = 0, x = 6, x = -6 x 2 = 0 x – 6 = 0 x + 6 = 0

3. Tom Worthing Intermediate Algebra All rights reserved. 3 Ex. 2 Solve x = -2 x = 2 x = ± 3i x 2 + 9 = 0 x – 2 = 0 x + 2 = 0 x 2 = – 9 x – 2 = 0 x + 2 = 0

4. Tom Worthing Intermediate Algebra All rights reserved. 4 Radical Equations An equation that involves radical expressions 1.Isolate the radical. 2.Raise both sides to power of index. 3.If a radical still remains repeat 1-2 4.Simplify and arrange terms 5.Solve 6.Check for extraneous roots (occurs when raising to even powers)

5. Tom Worthing Intermediate Algebra All rights reserved. 5 x + 1 = 3x + 1 0 = 2x 0 = x Check: Ex. 3 Solve

6. Tom Worthing Intermediate Algebra All rights reserved. 6 Equations with Fractional Exponents Some equations may be solved by raising both sides to reciprocal power Check all solutions Ex. 4 x 2/3 = 4 (x 2/3 ) 3/2 = 4 3/2 x = 8 Raise both sides to 3/2 power

7. Tom Worthing Intermediate Algebra All rights reserved. 7 Other equations that are quadratic in form 1.x 4 – 8x 2 + 15 = 0 2.x 6 + x 3 – 12 = 0 Solve by substituting and then factoring or by the calculator

8. Tom Worthing Intermediate Algebra All rights reserved. 8 Ex. 5 Solve 2x 2/3 + 7x 1/3 – 4 = 0 Substitute u = x 1/3 2u 2 + 7u – 4 = 0 Factor (2u – 1)(u + 4) = 0 Solve 2u – 1 = 0 u + 4 = 0 u = 0.5 u = – 4 Substitute x 1/3 = u x 1/3 = 0.5 x 1/3 = – 4 Solve x = (0.5) 3 = 0.125 x = (– 4) 3 = – 64