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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, let's go through each step together.
1. Identify the Substitution:
Notice that the equation involves [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We can simplify the solving process by substituting [tex]\(u = x^2\)[/tex].
2. Rewrite the Equation Using the Substitution:
If we let [tex]\(u = x^2\)[/tex], then [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Substitute and Simplify:
Replace [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] in the original equation with [tex]\(u^2\)[/tex] and [tex]\(u\)[/tex] respectively:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0.
\][/tex]
4. Resulting Quadratic Equation:
The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the correct substitution to rewrite the given polynomial as a quadratic equation is [tex]\(u = x^2\)[/tex].
The answer is:
[tex]\[ u = x^2. \][/tex]
1. Identify the Substitution:
Notice that the equation involves [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. We can simplify the solving process by substituting [tex]\(u = x^2\)[/tex].
2. Rewrite the Equation Using the Substitution:
If we let [tex]\(u = x^2\)[/tex], then [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Substitute and Simplify:
Replace [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] in the original equation with [tex]\(u^2\)[/tex] and [tex]\(u\)[/tex] respectively:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0.
\][/tex]
4. Resulting Quadratic Equation:
The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the correct substitution to rewrite the given polynomial as a quadratic equation is [tex]\(u = x^2\)[/tex].
The answer is:
[tex]\[ u = x^2. \][/tex]
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