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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we want to use a substitution that simplifies it into a quadratic form. Let's go through the steps:
1. Identify the terms:
- The equation is in terms of [tex]\(x\)[/tex], with a highest degree of 4. This means it's a quartic equation because the highest power of [tex]\(x\)[/tex] is 4.
2. Choose a substitution:
- We want to eliminate the quartic nature and reduce the equation to a quadratic form. A helpful substitution is to let [tex]\(u = x^2\)[/tex].
3. Apply the substitution:
- If [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
4. Rewrite the original equation:
- Substitute [tex]\(u = x^2\)[/tex] into the equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex].
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
The equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
5. Result:
- Now you have a quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
Therefore, the correct substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify the terms:
- The equation is in terms of [tex]\(x\)[/tex], with a highest degree of 4. This means it's a quartic equation because the highest power of [tex]\(x\)[/tex] is 4.
2. Choose a substitution:
- We want to eliminate the quartic nature and reduce the equation to a quadratic form. A helpful substitution is to let [tex]\(u = x^2\)[/tex].
3. Apply the substitution:
- If [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
4. Rewrite the original equation:
- Substitute [tex]\(u = x^2\)[/tex] into the equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex].
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
The equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
5. Result:
- Now you have a quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
Therefore, the correct substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
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