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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to choose a substitution that simplifies the expression. Here's how we can do that step-by-step:
1. Recognize the structure of the equation: The equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] involves terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Consider possible substitutions: A useful substitution can be made by letting [tex]\(u = x^2\)[/tex]. This is a common approach when an equation includes both [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] terms.
3. Transform the equation: With [tex]\(u = x^2\)[/tex], note that:
- [tex]\(x^4\)[/tex] can be rewritten as [tex]\((x^2)^2 = u^2\)[/tex].
4. Substitute into the original equation: Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the equation:
- The term [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
- The term [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
5. Write the new quadratic equation: After substitution, the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, the substitution [tex]\(u = x^2\)[/tex] is the correct choice to rewrite the original equation as a quadratic equation in terms of [tex]\(u\)[/tex].
1. Recognize the structure of the equation: The equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] involves terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Consider possible substitutions: A useful substitution can be made by letting [tex]\(u = x^2\)[/tex]. This is a common approach when an equation includes both [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] terms.
3. Transform the equation: With [tex]\(u = x^2\)[/tex], note that:
- [tex]\(x^4\)[/tex] can be rewritten as [tex]\((x^2)^2 = u^2\)[/tex].
4. Substitute into the original equation: Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the equation:
- The term [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
- The term [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
5. Write the new quadratic equation: After substitution, the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, the substitution [tex]\(u = x^2\)[/tex] is the correct choice to rewrite the original equation as a quadratic equation in terms of [tex]\(u\)[/tex].
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