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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use substitution to simplify it.
1. Identify a suitable substitution:
Let's substitute [tex]\( u = x^2 \)[/tex]. This means that [tex]\( x^4 \)[/tex] can be rewritten as [tex]\((x^2)^2 = u^2\)[/tex].
2. Rewrite the equation using the substitution:
- The term [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex].
- The term [tex]\(x^2\)[/tex] remains as [tex]\(u\)[/tex].
Substitute these into the original equation:
[tex]\[
4(x^4) - 21(x^2) + 20 = 0
\][/tex]
becomes:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
3. Simplify the rewritten equation:
The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now in standard quadratic form, where [tex]\(4u^2\)[/tex] corresponds to the quadratic term, [tex]\(-21u\)[/tex] is the linear term, and [tex]\(20\)[/tex] is the constant term.
This substitution simplifies the original polynomial into a quadratic equation in terms of [tex]\(u\)[/tex], allowing it to be solved using methods appropriate for quadratic equations. Therefore, the correct substitution to rewrite the original polynomial as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify a suitable substitution:
Let's substitute [tex]\( u = x^2 \)[/tex]. This means that [tex]\( x^4 \)[/tex] can be rewritten as [tex]\((x^2)^2 = u^2\)[/tex].
2. Rewrite the equation using the substitution:
- The term [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex].
- The term [tex]\(x^2\)[/tex] remains as [tex]\(u\)[/tex].
Substitute these into the original equation:
[tex]\[
4(x^4) - 21(x^2) + 20 = 0
\][/tex]
becomes:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
3. Simplify the rewritten equation:
The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now in standard quadratic form, where [tex]\(4u^2\)[/tex] corresponds to the quadratic term, [tex]\(-21u\)[/tex] is the linear term, and [tex]\(20\)[/tex] is the constant term.
This substitution simplifies the original polynomial into a quadratic equation in terms of [tex]\(u\)[/tex], allowing it to be solved using methods appropriate for quadratic equations. Therefore, the correct substitution to rewrite the original polynomial as a quadratic equation is [tex]\(u = x^2\)[/tex].
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