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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a substitution that simplifies it into a quadratic form. Here's how you can do it step-by-step:
1. Identify the substitution: Notice that the equation is in terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We want to rewrite it using a single variable. Since [tex]\(x^4 = (x^2)^2\)[/tex], we can set [tex]\(u = x^2\)[/tex]. This implies that [tex]\(x^4 = u^2\)[/tex].
2. Substitute into the equation: Replace [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(u\)[/tex] respectively:
[tex]\[
4(x^4) - 21(x^2) + 20 = 0 \quad \Rightarrow \quad 4(u^2) - 21(u) + 20 = 0
\][/tex]
3. Rewrite the equation: Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is in the standard quadratic form [tex]\(au^2 + bu + c = 0\)[/tex].
Therefore, the correct substitution that transforms the original equation into a quadratic equation is [tex]\(u = x^2\)[/tex]. This substitution simplifies the problem, making it easier to apply methods for solving quadratic equations.
1. Identify the substitution: Notice that the equation is in terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We want to rewrite it using a single variable. Since [tex]\(x^4 = (x^2)^2\)[/tex], we can set [tex]\(u = x^2\)[/tex]. This implies that [tex]\(x^4 = u^2\)[/tex].
2. Substitute into the equation: Replace [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(u\)[/tex] respectively:
[tex]\[
4(x^4) - 21(x^2) + 20 = 0 \quad \Rightarrow \quad 4(u^2) - 21(u) + 20 = 0
\][/tex]
3. Rewrite the equation: Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is in the standard quadratic form [tex]\(au^2 + bu + c = 0\)[/tex].
Therefore, the correct substitution that transforms the original equation into a quadratic equation is [tex]\(u = x^2\)[/tex]. This substitution simplifies the problem, making it easier to apply methods for solving quadratic equations.
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