Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a substitution method. Here's how you can do it step by step:
1. Identify the structure of the equation: The equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice it includes terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose an appropriate substitution: To convert this into a quadratic form, let's substitute [tex]\(u = x^2\)[/tex]. This choice works because it allows us to replace every [tex]\(x^2\)[/tex] with a [tex]\(u\)[/tex].
3. Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]:
- Since [tex]\(u = x^2\)[/tex], then [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
4. Rewrite the equation:
- Substitute into the original equation: [tex]\(4(x^4) - 21(x^2) + 20 = 0\)[/tex].
- This becomes: [tex]\(4(u^2) - 21u + 20 = 0\)[/tex].
5. Resulting quadratic equation: The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now in quadratic form, where [tex]\(u = x^2\)[/tex].
Therefore, the correct substitution to turn the given polynomial into a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify the structure of the equation: The equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice it includes terms with [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
2. Choose an appropriate substitution: To convert this into a quadratic form, let's substitute [tex]\(u = x^2\)[/tex]. This choice works because it allows us to replace every [tex]\(x^2\)[/tex] with a [tex]\(u\)[/tex].
3. Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex]:
- Since [tex]\(u = x^2\)[/tex], then [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
4. Rewrite the equation:
- Substitute into the original equation: [tex]\(4(x^4) - 21(x^2) + 20 = 0\)[/tex].
- This becomes: [tex]\(4(u^2) - 21u + 20 = 0\)[/tex].
5. Resulting quadratic equation: The equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is now in quadratic form, where [tex]\(u = x^2\)[/tex].
Therefore, the correct substitution to turn the given polynomial into a quadratic equation is [tex]\(u = x^2\)[/tex].
