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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 a step-by-step explanation:
1. Identify the structure of the equation: The equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice that the terms involve powers of [tex]\(x\)[/tex] that are multiples of 2, specifically [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose a substitution: To transform this into a standard quadratic form, we can use the substitution [tex]\( u = x^2 \)[/tex]. This is because [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
3. Rewrite the equation: Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Using the substitution [tex]\(u = x^2\)[/tex], this becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Resulting quadratic equation: After substitution, the equation is [tex]\(4u^2 - 21u + 20 = 0\)[/tex], which is a standard quadratic equation in terms of [tex]\(u\)[/tex].
So, the correct substitution to use in order to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify the structure of the equation: The equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice that the terms involve powers of [tex]\(x\)[/tex] that are multiples of 2, specifically [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose a substitution: To transform this into a standard quadratic form, we can use the substitution [tex]\( u = x^2 \)[/tex]. This is because [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
3. Rewrite the equation: Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Using the substitution [tex]\(u = x^2\)[/tex], this becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Resulting quadratic equation: After substitution, the equation is [tex]\(4u^2 - 21u + 20 = 0\)[/tex], which is a standard quadratic equation in terms of [tex]\(u\)[/tex].
So, the correct substitution to use in order to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
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