Answer :
To determine what substitution can be used to rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to look for a way to simplify it.
### Step-by-Step Solution:
1. Identify a suitable substitution:
- We observe that the terms in the equation involve powers of [tex]\(x\)[/tex] that are multiples of 2. Specifically, the terms are [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
- To simplify the equation, we can use a substitution to express everything in terms of the same variable.
- Let [tex]\(u = x^2\)[/tex]. This choice is natural because it will turn the [tex]\(x^4\)[/tex] term into something we can express as a square of [tex]\(u\)[/tex].
2. Express [tex]\(x^4\)[/tex] in terms of [tex]\(u\)[/tex]:
- Since [tex]\(x^4 = (x^2)^2\)[/tex], and we have [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = u^2\)[/tex].
3. Substitute in the original equation:
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] and [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] in the original equation:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
4. Resulting Quadratic Equation:
- The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, by using the substitution [tex]\(u = x^2\)[/tex], we have successfully rewritten the original polynomial equation as a quadratic equation.
### Step-by-Step Solution:
1. Identify a suitable substitution:
- We observe that the terms in the equation involve powers of [tex]\(x\)[/tex] that are multiples of 2. Specifically, the terms are [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
- To simplify the equation, we can use a substitution to express everything in terms of the same variable.
- Let [tex]\(u = x^2\)[/tex]. This choice is natural because it will turn the [tex]\(x^4\)[/tex] term into something we can express as a square of [tex]\(u\)[/tex].
2. Express [tex]\(x^4\)[/tex] in terms of [tex]\(u\)[/tex]:
- Since [tex]\(x^4 = (x^2)^2\)[/tex], and we have [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = u^2\)[/tex].
3. Substitute in the original equation:
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] and [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] in the original equation:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
4. Resulting Quadratic Equation:
- The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, by using the substitution [tex]\(u = x^2\)[/tex], we have successfully rewritten the original polynomial equation as a quadratic equation.
