Answer :
To transform the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] into a quadratic form, we can use a substitution method. Here's a step-by-step guide:
1. Identify a Suitable Substitution:
- We want to express the equation in terms of a single variable that can reduce the degree of the polynomial from 4 to 2.
- Notice that [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex]. So, let's set [tex]\(u = x^2\)[/tex].
2. Rewrite the Original Equation:
- Substitute [tex]\(u = x^2\)[/tex] into the equation.
- This changes [tex]\(x^4\)[/tex] into [tex]\(u^2\)[/tex] because [tex]\((x^2)^2 = x^4\)[/tex], and thus [tex]\(u^2 = x^4\)[/tex].
3. Transform the Equation:
- Replace all instances in the original equation:
- The term [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
- The term [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant [tex]\(+20\)[/tex] remains unchanged.
- The transformed equation is [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
4. Conclusion:
- The chosen substitution [tex]\(u = x^2\)[/tex] successfully transforms the original fourth degree polynomial into a quadratic equation: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
The correct substitution to rewrite the given equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is [tex]\(u = x^2\)[/tex].
1. Identify a Suitable Substitution:
- We want to express the equation in terms of a single variable that can reduce the degree of the polynomial from 4 to 2.
- Notice that [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex]. So, let's set [tex]\(u = x^2\)[/tex].
2. Rewrite the Original Equation:
- Substitute [tex]\(u = x^2\)[/tex] into the equation.
- This changes [tex]\(x^4\)[/tex] into [tex]\(u^2\)[/tex] because [tex]\((x^2)^2 = x^4\)[/tex], and thus [tex]\(u^2 = x^4\)[/tex].
3. Transform the Equation:
- Replace all instances in the original equation:
- The term [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
- The term [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant [tex]\(+20\)[/tex] remains unchanged.
- The transformed equation is [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
4. Conclusion:
- The chosen substitution [tex]\(u = x^2\)[/tex] successfully transforms the original fourth degree polynomial into a quadratic equation: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
The correct substitution to rewrite the given equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is [tex]\(u = x^2\)[/tex].
