To rewrite the polynomial equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a strategic substitution. Let's go through the process step-by-step:
1. Identify the Original Equation:
We are given the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex].
2. Consider a Suitable Substitution:
To reduce the degree of the polynomial, we set a suitable substitution. Notice that [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex]. Hence, we can let:
[tex]\[
u = x^2
\][/tex]
3. Rewrite the Equation in Terms of [tex]\(u\)[/tex]:
Substitute [tex]\(u = x^2\)[/tex] into the equation:
[tex]\[
x^4 = (x^2)^2 = u^2
\][/tex]
So the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] can be rewritten as:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Forming the Quadratic Equation:
After substituting [tex]\(u = x^2\)[/tex], the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is now a standard quadratic equation in terms of [tex]\(u\)[/tex].
5. Verify the Substitution Options:
Let's go through the given choices to determine the correct substitution:
- [tex]\(u = x^2\)[/tex]
- [tex]\(u = 2x^2\)[/tex]
- [tex]\(u = x^4\)[/tex]
- [tex]\(u = 4x^4\)[/tex]
Among these, the substitution that correctly transforms [tex]\(4x^4 - 21x^2 + 20\)[/tex] into the quadratic form [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is:
[tex]\[
u = x^2
\][/tex]
Hence, the correct substitution to rewrite [tex]\(4 x^4 - 21 x^2 + 20 = 0\)[/tex] as a quadratic equation is [tex]\(u = x^2\)[/tex], corresponding to choice number 1.
