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Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to make a substitution that simplifies the expression.
Here's a step-by-step method to achieve this:
1. Identify the terms:
- The given equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex].
2. Look for a substitution:
- Notice that the equation contains terms involving [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
- We can utilize a substitution that simplifies the powers of [tex]\(x\)[/tex].
3. Choose a suitable substitution:
- Let [tex]\(u = x^2\)[/tex]. This substitution is suitable because:
- [tex]\(u^2 = (x^2)^2 = x^4\)[/tex].
- Substitute into the original equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
4. Rewrite the equation:
- The original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
- This is now a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the substitution that should be used is [tex]\(u = x^2\)[/tex]. This turns the equation into a quadratic equation, making it simpler to solve for the roots using standard methods for quadratic equations.
Here's a step-by-step method to achieve this:
1. Identify the terms:
- The given equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex].
2. Look for a substitution:
- Notice that the equation contains terms involving [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
- We can utilize a substitution that simplifies the powers of [tex]\(x\)[/tex].
3. Choose a suitable substitution:
- Let [tex]\(u = x^2\)[/tex]. This substitution is suitable because:
- [tex]\(u^2 = (x^2)^2 = x^4\)[/tex].
- Substitute into the original equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
4. Rewrite the equation:
- The original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] becomes:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]
- This is now a quadratic equation in terms of [tex]\(u\)[/tex].
Thus, the substitution that should be used is [tex]\(u = x^2\)[/tex]. This turns the equation into a quadratic equation, making it simpler to solve for the roots using standard methods for quadratic equations.
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