Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, follow these steps:
1. Identify the Target Form: We want to transform the given equation into a standard quadratic form, which is [tex]\(au^2 + bu + c = 0\)[/tex], where [tex]\(u\)[/tex] is some expression involving [tex]\(x\)[/tex].
2. Choose the Substitution: Notice that the given equation has terms [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. If we let [tex]\(u = x^2\)[/tex], then:
- [tex]\(x^4\)[/tex] can be rewritten as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
3. Rewrite the Equation: Substitute [tex]\(u = x^2\)[/tex] into the given equation.
- [tex]\(4x^4\)[/tex] becomes [tex]\(4(u^2)\)[/tex].
- [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
Now, the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] transforms into:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Result: The equation is now a quadratic in [tex]\(u\)[/tex]: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
The appropriate substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify the Target Form: We want to transform the given equation into a standard quadratic form, which is [tex]\(au^2 + bu + c = 0\)[/tex], where [tex]\(u\)[/tex] is some expression involving [tex]\(x\)[/tex].
2. Choose the Substitution: Notice that the given equation has terms [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. If we let [tex]\(u = x^2\)[/tex], then:
- [tex]\(x^4\)[/tex] can be rewritten as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
3. Rewrite the Equation: Substitute [tex]\(u = x^2\)[/tex] into the given equation.
- [tex]\(4x^4\)[/tex] becomes [tex]\(4(u^2)\)[/tex].
- [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
Now, the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] transforms into:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Result: The equation is now a quadratic in [tex]\(u\)[/tex]: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
The appropriate substitution to rewrite the original equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
