Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, you can use a substitution method. Here's how to do it step-by-step:
1. Identify a substitution:
To simplify the given quartic equation (an equation of degree 4), we can use a substitution. Notice that the equation involves [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. A helpful substitution is [tex]\(u = x^2\)[/tex]. This substitution works because [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2 = u^2\)[/tex].
2. Apply the substitution:
Replace every instance of [tex]\(x^2\)[/tex] in the original equation with [tex]\(u\)[/tex]:
- The [tex]\(x^4\)[/tex] term becomes [tex]\(u^2\)[/tex] because [tex]\(x^4 = (x^2)^2\)[/tex].
- The [tex]\(x^2\)[/tex] term becomes [tex]\(u\)[/tex].
So, the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] transforms into:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
3. Resulting equation:
After substitution, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
By using the substitution [tex]\(u = x^2\)[/tex], the original quartic equation is rewritten as a quadratic equation, making it easier to solve for [tex]\(u\)[/tex] using standard methods for solving quadratic equations. After solving for [tex]\(u\)[/tex], substitute back to find [tex]\(x\)[/tex] by solving [tex]\(u = x^2\)[/tex].
1. Identify a substitution:
To simplify the given quartic equation (an equation of degree 4), we can use a substitution. Notice that the equation involves [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex]. A helpful substitution is [tex]\(u = x^2\)[/tex]. This substitution works because [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2 = u^2\)[/tex].
2. Apply the substitution:
Replace every instance of [tex]\(x^2\)[/tex] in the original equation with [tex]\(u\)[/tex]:
- The [tex]\(x^4\)[/tex] term becomes [tex]\(u^2\)[/tex] because [tex]\(x^4 = (x^2)^2\)[/tex].
- The [tex]\(x^2\)[/tex] term becomes [tex]\(u\)[/tex].
So, the original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] transforms into:
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]
3. Resulting equation:
After substitution, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\(u\)[/tex].
By using the substitution [tex]\(u = x^2\)[/tex], the original quartic equation is rewritten as a quadratic equation, making it easier to solve for [tex]\(u\)[/tex] using standard methods for solving quadratic equations. After solving for [tex]\(u\)[/tex], substitute back to find [tex]\(x\)[/tex] by solving [tex]\(u = x^2\)[/tex].
