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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to use a substitution that simplifies the equation into a standard quadratic form.
1. Identify the structure of the polynomial:
Notice that the polynomial includes terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. Specifically, [tex]\(x^4\)[/tex] is a power that is twice the power of [tex]\(x^2\)[/tex].
2. Choose an appropriate substitution:
We can use the substitution [tex]\(u = x^2\)[/tex]. This is a common technique when dealing with equations where one exponent is twice the other, like [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
3. Rewrite each term using the substitution:
- The term [tex]\(x^4\)[/tex] can be rewritten as [tex]\((x^2)^2\)[/tex], which, using our substitution [tex]\(u = x^2\)[/tex], becomes [tex]\(u^2\)[/tex].
- The term [tex]\(x^2\)[/tex] directly becomes [tex]\(u\)[/tex].
4. Substitute into the equation:
Original equation: [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]
Applying the substitution, we get:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the substitution [tex]\(u = x^2\)[/tex] should be used to rewrite the equation as a quadratic in [tex]\(u\)[/tex].
1. Identify the structure of the polynomial:
Notice that the polynomial includes terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. Specifically, [tex]\(x^4\)[/tex] is a power that is twice the power of [tex]\(x^2\)[/tex].
2. Choose an appropriate substitution:
We can use the substitution [tex]\(u = x^2\)[/tex]. This is a common technique when dealing with equations where one exponent is twice the other, like [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
3. Rewrite each term using the substitution:
- The term [tex]\(x^4\)[/tex] can be rewritten as [tex]\((x^2)^2\)[/tex], which, using our substitution [tex]\(u = x^2\)[/tex], becomes [tex]\(u^2\)[/tex].
- The term [tex]\(x^2\)[/tex] directly becomes [tex]\(u\)[/tex].
4. Substitute into the equation:
Original equation: [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]
Applying the substitution, we get:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the substitution [tex]\(u = x^2\)[/tex] should be used to rewrite the equation as a quadratic in [tex]\(u\)[/tex].
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