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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we should use a substitution method. Let's go through the steps together:
1. Identify the substitution needed:
Since the equation involves terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex], we can make a substitution to simplify it into a quadratic form. The key is to express the higher degree term in terms of a new variable.
2. Choose the substitution:
Let's set [tex]\(u = x^2\)[/tex]. This means:
- [tex]\(x^4\)[/tex] becomes [tex]\( (x^2)^2\)[/tex], which simplifies to [tex]\(u^2\)[/tex].
- [tex]\(x^2\)[/tex] becomes [tex]\(u\)[/tex].
3. Rewrite the equation:
Substitute [tex]\(u = x^2\)[/tex] into the original equation:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Conclusion:
The substitution [tex]\(u = x^2\)[/tex] successfully transforms the original equation into the quadratic equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
Therefore, the correct substitution to rewrite the equation as a quadratic is [tex]\(u = x^2\)[/tex].
1. Identify the substitution needed:
Since the equation involves terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex], we can make a substitution to simplify it into a quadratic form. The key is to express the higher degree term in terms of a new variable.
2. Choose the substitution:
Let's set [tex]\(u = x^2\)[/tex]. This means:
- [tex]\(x^4\)[/tex] becomes [tex]\( (x^2)^2\)[/tex], which simplifies to [tex]\(u^2\)[/tex].
- [tex]\(x^2\)[/tex] becomes [tex]\(u\)[/tex].
3. Rewrite the equation:
Substitute [tex]\(u = x^2\)[/tex] into the original equation:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Conclusion:
The substitution [tex]\(u = x^2\)[/tex] successfully transforms the original equation into the quadratic equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
Therefore, the correct substitution to rewrite the equation as a quadratic is [tex]\(u = x^2\)[/tex].
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