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Answer :
To solve the problem of rewriting the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to find a substitution that simplifies the equation into a standard quadratic form.
1. Identify the Form of the Equation: The given polynomial is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. This involves terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose an Appropriate Substitution: We are looking for a transformation that can reduce this polynomial to the form of [tex]\(ax^2 + bx + c = 0\)[/tex].
3. Substitution for Simplification: Let's use [tex]\(u = x^2\)[/tex]. This choice is based on:
- If [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\((x^2)^2 = u^2\)[/tex].
4. Rewrite the Equation: Substitute [tex]\(u\)[/tex] into the equation:
- [tex]\(4(x^4) = 4(u^2)\)[/tex]
- Therefore, the equation becomes [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
5. Resulting Quadratic Equation: With the substitution [tex]\(u = x^2\)[/tex], the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] is transformed into the quadratic equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
Thus, the correct substitution to rewrite the given equation as a quadratic is [tex]\(u = x^2\)[/tex].
1. Identify the Form of the Equation: The given polynomial is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. This involves terms of [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose an Appropriate Substitution: We are looking for a transformation that can reduce this polynomial to the form of [tex]\(ax^2 + bx + c = 0\)[/tex].
3. Substitution for Simplification: Let's use [tex]\(u = x^2\)[/tex]. This choice is based on:
- If [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\((x^2)^2 = u^2\)[/tex].
4. Rewrite the Equation: Substitute [tex]\(u\)[/tex] into the equation:
- [tex]\(4(x^4) = 4(u^2)\)[/tex]
- Therefore, the equation becomes [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
5. Resulting Quadratic Equation: With the substitution [tex]\(u = x^2\)[/tex], the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] is transformed into the quadratic equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
Thus, the correct substitution to rewrite the given equation as a quadratic is [tex]\(u = x^2\)[/tex].
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