answer
![x = \pm 2\sqrt{2} \: \: or \: \: x = \pm 1](/tex.php?f=x = \pm 2\sqrt{2} \: \: or \: \: x = \pm 1)
explanation
the given equation is
![{x}^{4} - 9 {x}^{2} + 8 = 0](/tex.php?f= {x}^{4} - 9 {x}^{2} + 8 = 0)
we can rewrite this as:
![({x}^{2})^{2} - 9 {x}^{2} + 8 = 0](/tex.php?f=({x}^{2})^{2} - 9 {x}^{2} + 8 = 0)
let
![u = {x}^{2}](/tex.php?f=u = {x}^{2} )
then the equation becomes:
![{u}^{2} - 9 {u}^{2} + 8 = 0](/tex.php?f= {u}^{2} - 9 {u}^{2} + 8 = 0)
the factors of 8 that sum up to -9 are {-1,-8}
we split the middle term with these factors to obtain,
![{u}^{2} - u - 8u+ 8 = 0](/tex.php?f= {u}^{2} - u - 8u+ 8 = 0)
we factor by grouping to obtain:
![u(u - 1) - 8(u - 1) = 0](/tex.php?f=u(u - 1) - 8(u - 1) = 0)
![(u - 8)(u - 1) = 0](/tex.php?f=(u - 8)(u - 1) = 0)
apply the zero product property to get:
![u - 8 = 0 \: \: or \: \: u - 1 = 0](/tex.php?f=u - 8 = 0 \: \: or \: \: u - 1 = 0)
![u = 8 \: or \: u = 1](/tex.php?f=u = 8 \: or \: u = 1)
but
![u = {x}^{2}](/tex.php?f=u = {x}^{2} )
![{x}^{2} = 8 \: or \: {x}^{2} = 1](/tex.php?f= {x}^{2} = 8 \: or \: {x}^{2} = 1)
![x = \pm \sqrt{8} \: \: or \: \: x = \pm \: \sqrt{ 1}](/tex.php?f=x = \pm \sqrt{8} \: \: or \: \: x = \pm \: \sqrt{ 1} )
![x = \pm 2\sqrt{2} \: \: or \: \: x = \pm 1](/tex.php?f=x = \pm 2\sqrt{2} \: \: or \: \: x = \pm 1)