Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a substitution method to simplify it.
### Step-by-step Solution:
1. Identify the substitution:
We are working with an equation involving [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. To simplify it, we use the substitution [tex]\(u = x^2\)[/tex].
2. Rewrite the terms in terms of [tex]\(u\)[/tex]:
- Since [tex]\(u = x^2\)[/tex], we have [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Substitute into the original equation:
Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the equation:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Result:
The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now a quadratic equation in terms of [tex]\(u\)[/tex].
In conclusion, the substitution that should be used to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex]. This transformation simplifies the quartic equation into a standard quadratic form.
### Step-by-step Solution:
1. Identify the substitution:
We are working with an equation involving [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. To simplify it, we use the substitution [tex]\(u = x^2\)[/tex].
2. Rewrite the terms in terms of [tex]\(u\)[/tex]:
- Since [tex]\(u = x^2\)[/tex], we have [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Substitute into the original equation:
Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the equation:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Result:
The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now a quadratic equation in terms of [tex]\(u\)[/tex].
In conclusion, the substitution that should be used to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex]. This transformation simplifies the quartic equation into a standard quadratic form.
