Answer :
To rewrite [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to find a substitution that simplifies this equation into the standard quadratic form.
Let's consider the substitution [tex]\(u = x^2\)[/tex]. With this substitution, [tex]\(u^2 = (x^2)^2 = x^4\)[/tex].
Now let's substitute [tex]\(u\)[/tex] into the original equation:
1. Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex]. So, [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
2. Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]. So, [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
Substituting these back into the equation, we have:
[tex]\[ 4u^2 - 21u + 20 = 0. \][/tex]
This is now a quadratic equation in terms of [tex]\(u\)[/tex]. Therefore, the substitution that should be used is [tex]\(u = x^2\)[/tex].
The correct substitution to rewrite [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is: [tex]\( u = x^2 \)[/tex].
Let's consider the substitution [tex]\(u = x^2\)[/tex]. With this substitution, [tex]\(u^2 = (x^2)^2 = x^4\)[/tex].
Now let's substitute [tex]\(u\)[/tex] into the original equation:
1. Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex]. So, [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
2. Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]. So, [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
Substituting these back into the equation, we have:
[tex]\[ 4u^2 - 21u + 20 = 0. \][/tex]
This is now a quadratic equation in terms of [tex]\(u\)[/tex]. Therefore, the substitution that should be used is [tex]\(u = x^2\)[/tex].
The correct substitution to rewrite [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is: [tex]\( u = x^2 \)[/tex].
