Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation using substitution, we can set [tex]\(u = x^2\)[/tex].
Here's how it works step-by-step:
1. Identify an appropriate substitution: Recognize that [tex]\(x^4\)[/tex] can be expressed in terms of [tex]\(x^2\)[/tex]. Since [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\((x^2)^2 = u^2\)[/tex].
2. Substitute [tex]\(u\)[/tex] into the equation: Replace every [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] in the original equation with [tex]\(u\)[/tex] and [tex]\(u^2\)[/tex], respectively. The equation becomes:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Replacing using [tex]\(u = x^2\)[/tex], we have:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
3. Check the form of the new equation: Now the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a standard quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the substitution [tex]\(u = x^2\)[/tex] is correct, and it allows us to express and solve the original polynomial in the simpler quadratic form.
Here's how it works step-by-step:
1. Identify an appropriate substitution: Recognize that [tex]\(x^4\)[/tex] can be expressed in terms of [tex]\(x^2\)[/tex]. Since [tex]\(u = x^2\)[/tex], then [tex]\(x^4\)[/tex] becomes [tex]\((x^2)^2 = u^2\)[/tex].
2. Substitute [tex]\(u\)[/tex] into the equation: Replace every [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] in the original equation with [tex]\(u\)[/tex] and [tex]\(u^2\)[/tex], respectively. The equation becomes:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
Replacing using [tex]\(u = x^2\)[/tex], we have:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
3. Check the form of the new equation: Now the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a standard quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the substitution [tex]\(u = x^2\)[/tex] is correct, and it allows us to express and solve the original polynomial in the simpler quadratic form.
