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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to make a substitution that simplifies the expression into a standard quadratic form.
Let's follow these steps:
1. Identify the Terms: Notice that the equation is in terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We need to express it in terms of a new variable so that it becomes a quadratic equation.
2. Choose a Substitution: We want to choose a substitution that relates to the lower-order term, which is [tex]\(x^2\)[/tex]. Let’s substitute:
[tex]\[
u = x^2
\][/tex]
With this substitution, [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex] (because [tex]\((x^2)^2 = x^4\)[/tex]).
3. Rewrite the Equation: Replace [tex]\(x^2\)[/tex] and [tex]\(x^4\)[/tex] in the original equation with [tex]\(u\)[/tex] and [tex]\(u^2\)[/tex] respectively:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
Becomes:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
4. Resulting Equation: Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
By making the substitution [tex]\(u = x^2\)[/tex], we've successfully rewritten the original equation as a quadratic equation. Therefore, the correct substitution is [tex]\(u = x^2\)[/tex].
Let's follow these steps:
1. Identify the Terms: Notice that the equation is in terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We need to express it in terms of a new variable so that it becomes a quadratic equation.
2. Choose a Substitution: We want to choose a substitution that relates to the lower-order term, which is [tex]\(x^2\)[/tex]. Let’s substitute:
[tex]\[
u = x^2
\][/tex]
With this substitution, [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex] (because [tex]\((x^2)^2 = x^4\)[/tex]).
3. Rewrite the Equation: Replace [tex]\(x^2\)[/tex] and [tex]\(x^4\)[/tex] in the original equation with [tex]\(u\)[/tex] and [tex]\(u^2\)[/tex] respectively:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
Becomes:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
4. Resulting Equation: Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
By making the substitution [tex]\(u = x^2\)[/tex], we've successfully rewritten the original equation as a quadratic equation. Therefore, the correct substitution is [tex]\(u = x^2\)[/tex].
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