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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to perform a substitution that transforms the equation into a standard quadratic form.
Here’s how we can do it:
1. Identify terms involving [tex]\(x\)[/tex]: Notice that this equation involves powers of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. Specifically, [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex].
2. Choose an appropriate substitution: To simplify the equation to a quadratic form, let’s set [tex]\(u = x^2\)[/tex]. This substitution turns the term [tex]\(x^4\)[/tex] into [tex]\(u^2\)[/tex] because [tex]\((x^2)^2 = u^2\)[/tex].
3. Substitute and simplify: Replace all occurrences of [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation. The equation becomes:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Substitute [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
Now, the original equation is rewritten as the quadratic equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] in terms of [tex]\(u\)[/tex].
Therefore, the correct substitution to use for rewriting the given equation as a quadratic one is [tex]\(u = x^2\)[/tex].
Here’s how we can do it:
1. Identify terms involving [tex]\(x\)[/tex]: Notice that this equation involves powers of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. Specifically, [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex].
2. Choose an appropriate substitution: To simplify the equation to a quadratic form, let’s set [tex]\(u = x^2\)[/tex]. This substitution turns the term [tex]\(x^4\)[/tex] into [tex]\(u^2\)[/tex] because [tex]\((x^2)^2 = u^2\)[/tex].
3. Substitute and simplify: Replace all occurrences of [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation. The equation becomes:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Substitute [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
Now, the original equation is rewritten as the quadratic equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] in terms of [tex]\(u\)[/tex].
Therefore, the correct substitution to use for rewriting the given equation as a quadratic one is [tex]\(u = x^2\)[/tex].
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