Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to use a substitution that simplifies the higher power terms to a quadratic form. Here's how you can do it:
1. Identify the relationship in terms of [tex]\(x^2\)[/tex]:
- Notice that [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex], i.e., [tex]\(x^4 = (x^2)^2\)[/tex].
2. Choose the substitution:
- Let [tex]\(u = x^2\)[/tex]. This simplifies the expression since [tex]\(x^4 = u^2\)[/tex].
3. Substitute [tex]\(u\)[/tex] into the original equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] using the substitution [tex]\(u = x^2\)[/tex].
- The original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] now becomes:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
- Simplify the equation:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
Now the equation is a quadratic in terms of [tex]\(u\)[/tex], and the correct substitution to achieve this transformation is [tex]\(u = x^2\)[/tex]. This substitution helps in solving the original equation by turning it into a more familiar quadratic form.
1. Identify the relationship in terms of [tex]\(x^2\)[/tex]:
- Notice that [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex], i.e., [tex]\(x^4 = (x^2)^2\)[/tex].
2. Choose the substitution:
- Let [tex]\(u = x^2\)[/tex]. This simplifies the expression since [tex]\(x^4 = u^2\)[/tex].
3. Substitute [tex]\(u\)[/tex] into the original equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] using the substitution [tex]\(u = x^2\)[/tex].
- The original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] now becomes:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
- Simplify the equation:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
Now the equation is a quadratic in terms of [tex]\(u\)[/tex], and the correct substitution to achieve this transformation is [tex]\(u = x^2\)[/tex]. This substitution helps in solving the original equation by turning it into a more familiar quadratic form.
