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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to find an appropriate substitution that will simplify the original equation.
Let's look at the expression carefully:
1. Notice the term [tex]\(4x^4\)[/tex]. This can be expressed as [tex]\((x^2)^2\)[/tex], because [tex]\(x^4 = (x^2)^2\)[/tex].
2. The given substitution options are:
- [tex]\(u = x^2\)[/tex]
- [tex]\(u = 2x^2\)[/tex]
- [tex]\(u = x^4\)[/tex]
- [tex]\(u = 4x^4\)[/tex]
3. To transform the equation into a quadratic in terms of a new variable [tex]\(u\)[/tex], we need to substitute [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]. This makes sense since the term [tex]\(4x^4\)[/tex] turns into [tex]\(4(x^2)^2 = 4u^2\)[/tex].
4. Applying this substitution throughout the equation:
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex], turning [tex]\(4x^4\)[/tex] into [tex]\(4u^2\)[/tex].
- The term [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant [tex]\(20\)[/tex] remains the same.
5. Therefore, the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is now a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the correct substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
Let's look at the expression carefully:
1. Notice the term [tex]\(4x^4\)[/tex]. This can be expressed as [tex]\((x^2)^2\)[/tex], because [tex]\(x^4 = (x^2)^2\)[/tex].
2. The given substitution options are:
- [tex]\(u = x^2\)[/tex]
- [tex]\(u = 2x^2\)[/tex]
- [tex]\(u = x^4\)[/tex]
- [tex]\(u = 4x^4\)[/tex]
3. To transform the equation into a quadratic in terms of a new variable [tex]\(u\)[/tex], we need to substitute [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]. This makes sense since the term [tex]\(4x^4\)[/tex] turns into [tex]\(4(x^2)^2 = 4u^2\)[/tex].
4. Applying this substitution throughout the equation:
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex], turning [tex]\(4x^4\)[/tex] into [tex]\(4u^2\)[/tex].
- The term [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant [tex]\(20\)[/tex] remains the same.
5. Therefore, the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is now a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the correct substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
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