To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a suitable substitution. Let's go through the steps to determine the appropriate substitution step-by-step.
1. Identify the target form: We need to transform the given equation into a standard quadratic form, which is generally [tex]\(au^2 + bu + c = 0\)[/tex].
2. Analyze the equation: The given equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice the terms involve [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
3. Choose a substitution: We want to transform the variable so that [tex]\(x^4\)[/tex] becomes a quadratic term and [tex]\(x^2\)[/tex] becomes a linear term.
- If we let [tex]\(u = x^2\)[/tex], then:
[tex]\[ x^4 = (x^2)^2 = u^2 \][/tex]
Thus, substituting [tex]\(u\)[/tex] into the equation, we get:
[tex]\[ 4(u^2) - 21u + 20 = 0 \][/tex]
This simplifies to:
[tex]\[ 4u^2 - 21u + 20 = 0 \][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
- Consider other options:
- [tex]\(u = 2x^2\)[/tex]: This would make the substitution more complex and would not lead to a straightforward quadratic equation.
- [tex]\(u = x^4\)[/tex]: This does not simplify our goal, as [tex]\(u = x^4\)[/tex] would not eliminate the [tex]\(x^2\)[/tex] term properly.
- [tex]\(u = 4x^4\)[/tex]: Similar to [tex]\(u = x^4\)[/tex], it would not simplify the equation to the desired quadratic form.
4. Conclusion: The correct substitution that rewrites the given equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is [tex]\(u = x^2\)[/tex].
The correct choice is:
[tex]\[ \boxed{u = x^2} \][/tex]
