Answer:
[tex](x ^ 2 -1)(x ^ 2 -8)\\\\x = 2\sqrt{2}\\\\x = - 2\sqrt{2}\\\\x = 1\\\\x = -1[/tex]
Step-by-step explanation:
We have the following equation [tex]x^4 - 9x2 + 8 = 0[/tex]
To solve this problem, do:
[tex]u = x ^ 2[/tex]
then it has:
[tex]u ^ 2 - 9u + 8 = 0[/tex]
Then he has a second-degree equation.
To resolve factor.
You must find 2 numbers that when you add them get -9 and multiply them get 8.
These numbers are:
u = -1 and u = -8
Then it has:
[tex](u-1)(u-8) = 0[/tex]
[tex]u = 1[/tex]
and
[tex]u = 8[/tex]
Remember that [tex]u = x ^ 2[/tex]
Then it has:
[tex](x ^ 2 -1)(x ^ 2 -8)\\\\x = 2\sqrt{2}\\\\x = - 2\sqrt{2}\\\\x = 1\\\\x = -1[/tex]
ANSWER
[tex]x = \pm 2\sqrt{2} \: or \: {x}= \pm1[/tex]
EXPLANATION
Given;
[tex] {x}^{4} - 9 {x}^{2} +8 = 0[/tex]
We can rewrite as
[tex] ({x}^{2} )^{2} - 9 {x}^{2} +8 = 0[/tex]
Let
[tex]u = {x}^{2} [/tex]
This implies that,
[tex] u^{2} - 9u +8 = 0[/tex]
This implies that,
[tex] u^{2}-u - 8u +8 = 0[/tex]
[tex]u(u - 1) - 8(u - 1) = 0[/tex]
[tex](u - 1)(u - 8) = 0[/tex]
[tex]u = 8 \: or \: u = 1[/tex]
But
[tex]u = {x}^{2} [/tex]
This implies that,
[tex] {x}^{2} = 8 \: or \: {x}^{2} = 1[/tex]
[tex] x = \pm \sqrt{8} \: or \: {x}= \pm \sqrt{1} [/tex]
[tex] x = \pm 2\sqrt{2} \: or \: {x}= \pm1[/tex]
