To find the substitution that transforms the original equation [tex]\(x^8 - 3x^4 + 2 = 0\)[/tex] into a quadratic equation, let's examine each option step-by-step.
### Option 1: [tex]\( u = x^2 \)[/tex]
Substituting [tex]\( u = x^2 \)[/tex] into the equation:
[tex]\[ x^8 - 3 x^4 + 2 = 0 \][/tex]
means revising the powers of [tex]\( x \)[/tex]:
[tex]\[ (u^4) - 3(u^2) + 2 = 0 \][/tex]
which transforms into:
[tex]\[ u^4 - 3u^2 + 2 = 0 \][/tex]
This is not a quadratic equation because the highest power of [tex]\( u \)[/tex] is 4.
### Option 2: [tex]\( u = x^4 \)[/tex]
Substituting [tex]\( u = x^4 \)[/tex] into the equation:
[tex]\[ x^8 - 3 x^4 + 2 = 0 \][/tex]
transforms it to:
[tex]\[ u^2 - 3u + 2 = 0 \][/tex]
This simplifies to:
[tex]\[ u^2 - 3u + 2 = 0 \][/tex]
This is a quadratic equation because the highest power of [tex]\( u \)[/tex] is 2.
### Option 3: [tex]\( u = x^8 \)[/tex]
Substituting [tex]\( u = x^8 \)[/tex] into the equation:
[tex]\[ x^8 - 3 x^4 + 2 = 0 \][/tex]
transforms it to:
[tex]\[ u - 3(x^4) + 2 = 0 \][/tex]
This does not transform into a quadratic form since the variable [tex]\( x^4 \)[/tex] remains and the powers are mixed.
### Option 4: [tex]\( u = x^{16} \)[/tex]
Substituting [tex]\( u = x^{16} \)[/tex] into the equation:
[tex]\[ x^8 - 3 x^4 + 2 = 0 \][/tex]
attempts to transform it to terms with [tex]\( x \)[/tex]:
[tex]\[ (u^{1/2}) - 3(x^4) + 2 = 0 \][/tex]
This does not transform into a simpler quadratic form either.
Thus, the best substitution to transform the given equation [tex]\( x^8 - 3 x^4 + 2 = 0 \)[/tex] into a quadratic equation is:
[tex]\[ u = x^4 \][/tex]
Therefore, the correct substitution is:
[tex]\[ u = x^4 \][/tex]
