Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a substitution to make it easier to solve.
Here's how you can do it step-by-step:
1. Identify the substitution variable: Look for a substitution that will simplify the expression. Since [tex]\(x^4\)[/tex] is the highest power of [tex]\(x\)[/tex] in the equation, you can look at [tex]\(x^2\)[/tex] which is a square of [tex]\(x\)[/tex].
2. Make the substitution: Let [tex]\(u = x^2\)[/tex]. This changes the equation because every instance of [tex]\(x^2\)[/tex] is replaced with [tex]\(u\)[/tex].
3. Rewrite the equation:
- Since [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
4. Choose the correct substitution from the options:
- The correct substitution is [tex]\(u = x^2\)[/tex]. This simplifies the equation into a quadratic form.
So, the substitution that should be used is [tex]\(u = x^2\)[/tex].
Here's how you can do it step-by-step:
1. Identify the substitution variable: Look for a substitution that will simplify the expression. Since [tex]\(x^4\)[/tex] is the highest power of [tex]\(x\)[/tex] in the equation, you can look at [tex]\(x^2\)[/tex] which is a square of [tex]\(x\)[/tex].
2. Make the substitution: Let [tex]\(u = x^2\)[/tex]. This changes the equation because every instance of [tex]\(x^2\)[/tex] is replaced with [tex]\(u\)[/tex].
3. Rewrite the equation:
- Since [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
This is a quadratic equation in terms of [tex]\(u\)[/tex].
4. Choose the correct substitution from the options:
- The correct substitution is [tex]\(u = x^2\)[/tex]. This simplifies the equation into a quadratic form.
So, the substitution that should be used is [tex]\(u = x^2\)[/tex].
