Respuesta :

TheBilogis
rewriting x^4 as A^2 and x^2 and A.

A^2-52A+576=0
(A-36)(A-16)=0
A=36,16

since A is x^2,

x^2=36 and x^2=16
x=6 and -6 and x=4 and -4
kudzordzifrancis
ANSWER

The solutions are,

[tex]x = - 6 \: or \: x = 6 \: or \: x = - 4 \: or \: x = 4[/tex]


EXPLANATION

The given polynomial equation is,

[tex] {x}^{4} - 52 {x}^{2} + 576 = 0[/tex]

We can rewrite the equation to obtain,



[tex]( {x}^{2} )^{2} - 52 {x}^{2} + 576 = 0[/tex]

If we let

[tex]u = {x}^{2} [/tex]

Then our equation becomes,


[tex] {u}^{2} - 52u + 576 = 0[/tex]


This is a quadratic equation that can be solved by factoring.


We split the middle term to obtain,


[tex] {u}^{2} - 16u - 36u+ 576 = 0[/tex]




This factors to give us,


[tex]u(u - 16) - 36(u - 16) = 0[/tex]


[tex](u - 16)(u - 36) = 0[/tex]

[tex]u = 16 \: or \: u = 36[/tex]

This implies that,

[tex] {x}^{2} = 16 \:or \: {x}^{2} = 36[/tex]

[tex]x = \pm4 \: or \: x = \pm6[/tex]

There real solutions are,

[tex]x = - 6 \: or \: x = 6 \: or \: x = - 4 \: or \: x = 4[/tex]

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