Equations » Higher degree equations

Example 1: with an odd expontent
Example 2: with an even exponent
Example 3: factorising (a power or) x
Example 4: factorising (a power or) x
Example 5: substitute x³ = p
Example 6: substitute x⁶ = p

Example 1

(2`x` – 1)⁵	= –3
2`x` – 1	=
2`x` – 1	= –
2`x`	= 1 –
`x`	= 12 – 12

Example 2

(3`x` – 6)⁴ = 10
3`x` – 6 = –	or 3`x` – 6 =
3`x` = 6 –	or 3`x` = 6 +
`x` = 2 – 13	or `x` = 2 + 13

Example 3

`x`³ – `x`² – 6`x`	= 0
`x` (`x`² – `x` – 6)	= 0
`x`(`x` – 3)(`x` + 2)	= 0

x = 0 or x = 3 or x = –2

Example 4

`x`⁵ – 2`x`⁴ – 8`x`³	= 0
`x`³(`x`² – 2`x` – 8)	= 0
`x`³(`x` – 4)(`x` + 2)	= 0

x = 0 or x = 4 or x = –2

Example 5

`x`⁶ – 6`x`³ + 8	= 0
Substitute: `p` = `x`³
`p`² – 6`p` + 8	= 0
(`p` – 2)(`p` – 4)	= 0

`p` = 2 or	`p` = 4
`x`³ = 2 or	`x`³ = 4
`x` = or	`x` =

Example 6

`x`¹² – `x`⁶ – 2	= 0
Substitute: `p` = `x`⁶
`p`² – `p` – 2	= 0
(`p` – 2)(`p` + 1)	= 0

`p` = 2 or	`p` = –1
`x`⁶ = 2 or	`x`⁶ = –1
`x` = – or	`x` =

Those are the only solutions as
x⁶ = –1 has no solutions.