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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] in a quadratic form, we can use substitution. Here's a step-by-step explanation:
1. Identify the Substitution: We want to convert this into a simpler quadratic equation. Notice that the term [tex]\(x^4\)[/tex] can be expressed in terms of [tex]\(x^2\)[/tex] if we use a substitution. If we let [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex].
2. Apply the Substitution:
- Substitute [tex]\(u = x^2\)[/tex] into the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex].
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] since [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
3. Rewrite the Equation in Quadratic Form:
Using the substitution, the equation becomes:
[tex]\[
4(u)^2 - 21(u) + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
So, the substitution [tex]\(u = x^2\)[/tex] simplifies the original polynomial equation into a quadratic form: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
1. Identify the Substitution: We want to convert this into a simpler quadratic equation. Notice that the term [tex]\(x^4\)[/tex] can be expressed in terms of [tex]\(x^2\)[/tex] if we use a substitution. If we let [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex].
2. Apply the Substitution:
- Substitute [tex]\(u = x^2\)[/tex] into the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex].
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] since [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
3. Rewrite the Equation in Quadratic Form:
Using the substitution, the equation becomes:
[tex]\[
4(u)^2 - 21(u) + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
So, the substitution [tex]\(u = x^2\)[/tex] simplifies the original polynomial equation into a quadratic form: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
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