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

Algebra ->  Quadratic Equations and Parabolas -> 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       Log On


   

Question 14028: Find all solutions of the equations.
1.
x6 - 64 = 0
2.
9x4 - 24x3 + 16x2 = 0

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!

Find all solutions of the equations.
1.
`
x6 - 64 = 0
`
First factor as the difference of squares.
`
(x3)2 - 82 = 0
`
(x3 - 8)(x3 + 8) = 0
`
Next factor these two parenthetical expressiona as the difference and
sum respectively of two cubes
`
(x3 - 23)(x3 + 23) = 0
`
(x - 2)(x2 + 2x + 4)(x + 2)(x2 - 2x + 4) = 0
`
Set each of the four factors = 0
`
x - 2 = 0, solving gives x = 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` _
x2 + 2x + 4 = 0, solving by the quadratic formula gives x = -1±Ö3
`
x + 2 = 0, solving gives x = -2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `_
x2 - 2x + 4 = 0, solving by the quadratic formula gives x = 1±Ö3
`
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`
2.
`
9x4 - 24x3 + 16x2 = 0
`
Factor out x2
`
x2(9x2 - 24x + 16) = 0
`
Factor the expression in parentheses
`
x2(3x - 4)(3x - 4) = 0
`
x2(3x -4)2 = 0
`
Set each factor equal to 0
`
x2 = 0 gives x = 0
`
(3x - 4)2 = 0, gives x = 4/3
`
Edwin