Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to make a suitable substitution that simplifies the form of the equation.
1. Identify the terms: The equation given is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex], where we have terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. Our goal is to turn this into a quadratic equation, which typically has the form [tex]\(au^2 + bu + c = 0\)[/tex].
2. Choose a substitution: We need to find a substitution that allows us to express the equation in a quadratic form. Notice:
- The term [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex].
3. Substitute [tex]\(u = x^2\)[/tex]:
- When [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\((x^2)^2 = u^2\)[/tex].
4. Rewrite the equation: Substitute [tex]\(u = x^2\)[/tex] into the original equation:
- [tex]\(4(x^4)\)[/tex] becomes [tex]\(4(u^2)\)[/tex].
- [tex]\(-21(x^2)\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant term [tex]\(+20\)[/tex] remains the same.
5. Form the quadratic equation: Now the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is a quadratic equation in the variable [tex]\(u\)[/tex].
Thus, the correct substitution to rewrite [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify the terms: The equation given is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex], where we have terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. Our goal is to turn this into a quadratic equation, which typically has the form [tex]\(au^2 + bu + c = 0\)[/tex].
2. Choose a substitution: We need to find a substitution that allows us to express the equation in a quadratic form. Notice:
- The term [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex].
3. Substitute [tex]\(u = x^2\)[/tex]:
- When [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\((x^2)^2 = u^2\)[/tex].
4. Rewrite the equation: Substitute [tex]\(u = x^2\)[/tex] into the original equation:
- [tex]\(4(x^4)\)[/tex] becomes [tex]\(4(u^2)\)[/tex].
- [tex]\(-21(x^2)\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant term [tex]\(+20\)[/tex] remains the same.
5. Form the quadratic equation: Now the equation becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is a quadratic equation in the variable [tex]\(u\)[/tex].
Thus, the correct substitution to rewrite [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is [tex]\(u = x^2\)[/tex].
