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
Tom Worthing Intermediate Algebra All rights reserved.
1
Solve x4 – 36x2 = 0 (4 possible)
Ex. 1
x 2 (x 2  36)  0
x  x  6  x  6  0
2
x2 = 0
x–6=0
x+6=0
x = 0, x = 0, x = 6, x = -6
Tom Worthing Intermediate Algebra All rights reserved.
2
Ex. 2
Solve x  5 x  36  0
4
x
x
2
2
2
 9  x  4   0
2
 9   x  2)( x  2   0
x2 + 9 = 0
x2 = – 9
x = -2
Tom Worthing Intermediate Algebra All rights reserved.
x–2=0
x–2=0
x=2
x+2=0
x+2=0
x = ± 3i
3
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)
Tom Worthing Intermediate Algebra All rights reserved.
4
Ex. 3
Solve
x  1  3x  1
( x  1)  ( 3 x  1)
2
2
x + 1 = 3x + 1
0 = 2x
0=x
Check:
0  1  3(0)  1
Tom Worthing Intermediate Algebra All rights reserved.
5
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
Raise both sides to 3/2 power
(x 2/3)3/2 = 4 3/2
x=8
Tom Worthing Intermediate Algebra All rights reserved.
6
Other equations that are quadratic in form
1. x4 – 8x2 + 15 = 0
2. x6 + x3 – 12 = 0
Solve by substituting and then factoring or
by the calculator
Tom Worthing Intermediate Algebra All rights reserved.
7
Ex. 5
2x2/3 + 7x1/3 – 4 = 0
Solve
2u2 + 7u – 4 = 0
Substitute u = x1/3
(2u – 1)(u + 4) = 0
Factor
2u – 1 = 0
Solve
u = 0.5
x1/3 = 0.5
u+4=0
u=–4
x1/3 = – 4
x = (0.5)3 = 0.125
Substitute x1/3 = u
Solve
x = (– 4)3 = – 64
Tom Worthing Intermediate Algebra All rights reserved.
8