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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we choose a suitable substitution.
Let's consider the substitution [tex]\(u = x^2\)[/tex].
Here's why this works:
1. Identify expressions with [tex]\(x\)[/tex]: In the original equation, we see terms [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Relate [tex]\(u\)[/tex] to these terms:
- Since [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Rewrite the equation: Substituting [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex], we get:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, the correct substitution to rewrite the original equation as a quadratic is [tex]\(u = x^2\)[/tex].
Let's consider the substitution [tex]\(u = x^2\)[/tex].
Here's why this works:
1. Identify expressions with [tex]\(x\)[/tex]: In the original equation, we see terms [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Relate [tex]\(u\)[/tex] to these terms:
- Since [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Rewrite the equation: Substituting [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex], we get:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
Therefore, the correct substitution to rewrite the original equation as a quadratic is [tex]\(u = x^2\)[/tex].
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