SOLUTION: Find all solutions of the equations. 1. x<sup>6</sup> - 64 = 0 2. 9x<sup>4</sup> - 24x<sup>3</sup> + 16x<sup>2</sup> = 0
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-> SOLUTION: Find all solutions of the equations. 1. x<sup>6</sup> - 64 = 0 2. 9x<sup>4</sup> - 24x<sup>3</sup> + 16x<sup>2</sup> = 0
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Question 14028
:
Find all solutions of the equations.
1.
x
6
- 64 = 0
2.
9x
4
- 24x
3
+ 16x
2
= 0
Answer by
AnlytcPhil(1810)
(
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Find all solutions of the equations.
1.
`
x
6
- 64 = 0
`
First factor as the difference of squares.
`
(x
3
)
2
- 8
2
= 0
`
(x
3
- 8)(x
3
+ 8) = 0
`
Next factor these two parenthetical expressiona as the difference and
sum respectively of two cubes
`
(x
3
- 2
3
)(x
3
+ 2
3
) = 0
`
(x - 2)(x
2
+ 2x + 4)(x + 2)(x
2
- 2x + 4) = 0
`
Set each of the four factors = 0
`
x - 2 = 0, solving gives x = 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` _
x
2
+ 2x + 4 = 0, solving by the quadratic formula gives x = -1±
Ö
3
`
x + 2 = 0, solving gives x = -2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `_
x
2
- 2x + 4 = 0, solving by the quadratic formula gives x = 1±
Ö
3
`
-----------------------------------------------
`
2.
`
9x
4
- 24x
3
+ 16x2 = 0
`
Factor out x
2
`
x
2
(9x
2
- 24x + 16) = 0
`
Factor the expression in parentheses
`
x
2
(3x - 4)(3x - 4) = 0
`
x
2
(3x -4)
2
= 0
`
Set each factor equal to 0
`
x
2
= 0 gives x = 0
`
(3x - 4)
2
= 0, gives x = 4/3
`
Edwin
