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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 approach it:
1. Identify Patterns:
- Notice that the equation involves powers of [tex]\(x\)[/tex]. The highest power is [tex]\(4\)[/tex], and the next highest power is [tex]\(2\)[/tex]. This suggests a substitution could simplify the problem to a quadratic form.
2. Choose the Substitution:
- Let's try substituting [tex]\(u = x^2\)[/tex]. This is because [tex]\(u = x^2\)[/tex] can help reduce the powers of [tex]\(x\)[/tex] to a simpler quadratic form. With this substitution, [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex] because [tex]\(x^4 = (x^2)^2\)[/tex].
3. Rewrite the Equation:
- Replace [tex]\(x^2\)[/tex] in the original equation with [tex]\(u\)[/tex]:
[tex]\[
4(x^2)^2 - 21x^2 + 20 = 0
\][/tex]
- Substituting [tex]\(u = x^2\)[/tex], it becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Result:
- You now have a quadratic equation in terms of [tex]\(u\)[/tex]: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
Therefore, the correct substitution to rewrite the original equation as a quadratic is [tex]\(u = x^2\)[/tex]. This substitution transforms the original form into a quadratic equation that can be solved using methods like factoring, the quadratic formula, or completing the square.
1. Identify Patterns:
- Notice that the equation involves powers of [tex]\(x\)[/tex]. The highest power is [tex]\(4\)[/tex], and the next highest power is [tex]\(2\)[/tex]. This suggests a substitution could simplify the problem to a quadratic form.
2. Choose the Substitution:
- Let's try substituting [tex]\(u = x^2\)[/tex]. This is because [tex]\(u = x^2\)[/tex] can help reduce the powers of [tex]\(x\)[/tex] to a simpler quadratic form. With this substitution, [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex] because [tex]\(x^4 = (x^2)^2\)[/tex].
3. Rewrite the Equation:
- Replace [tex]\(x^2\)[/tex] in the original equation with [tex]\(u\)[/tex]:
[tex]\[
4(x^2)^2 - 21x^2 + 20 = 0
\][/tex]
- Substituting [tex]\(u = x^2\)[/tex], it becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
4. Result:
- You now have a quadratic equation in terms of [tex]\(u\)[/tex]: [tex]\(4u^2 - 21u + 20 = 0\)[/tex].
Therefore, the correct substitution to rewrite the original equation as a quadratic is [tex]\(u = x^2\)[/tex]. This substitution transforms the original form into a quadratic equation that can be solved using methods like factoring, the quadratic formula, or completing the square.
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