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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 method. Here's the step-by-step explanation:
1. Identify the substitution:
We want to find a substitution that will transform the given equation into a quadratic form. Notice the powers of [tex]\(x\)[/tex] in the given polynomial: [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose the substitution:
Let [tex]\(u = x^2\)[/tex]. This is a suitable substitution because [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Rewrite the equation:
Substitute [tex]\(u = x^2\)[/tex] into the original equation. This gives:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Simplify this using the substitution:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This 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].
1. Identify the substitution:
We want to find a substitution that will transform the given equation into a quadratic form. Notice the powers of [tex]\(x\)[/tex] in the given polynomial: [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose the substitution:
Let [tex]\(u = x^2\)[/tex]. This is a suitable substitution because [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Rewrite the equation:
Substitute [tex]\(u = x^2\)[/tex] into the original equation. This gives:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Simplify this using the substitution:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This 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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