Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to find an appropriate substitution.
1. Observe the terms in the equation: the highest power of [tex]\(x\)[/tex] is 4, specifically in [tex]\(x^4\)[/tex], and another term involves [tex]\(x^2\)[/tex].
2. Notice that [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex].
3. This suggests a substitution, where we define a new variable:
[tex]\[
u = x^2
\][/tex]
4. Substitute [tex]\(u = x^2\)[/tex] into the equation:
- Since [tex]\(x^4 = (x^2)^2\)[/tex], then [tex]\(x^4 = u^2\)[/tex].
5. Replace each term of the original equation:
- [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex]
- [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex]
6. Substituting these into the equation, we have:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This transformed equation, [tex]\(4u^2 - 21u + 20 = 0\)[/tex], is now a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, the appropriate substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Observe the terms in the equation: the highest power of [tex]\(x\)[/tex] is 4, specifically in [tex]\(x^4\)[/tex], and another term involves [tex]\(x^2\)[/tex].
2. Notice that [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex].
3. This suggests a substitution, where we define a new variable:
[tex]\[
u = x^2
\][/tex]
4. Substitute [tex]\(u = x^2\)[/tex] into the equation:
- Since [tex]\(x^4 = (x^2)^2\)[/tex], then [tex]\(x^4 = u^2\)[/tex].
5. Replace each term of the original equation:
- [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex]
- [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex]
6. Substituting these into the equation, we have:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This transformed equation, [tex]\(4u^2 - 21u + 20 = 0\)[/tex], is now a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, the appropriate substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
