Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to find an appropriate substitution that simplifies the expression.
Let's examine the terms in the equation:
1. The term [tex]\(x^4\)[/tex] can be related to [tex]\(x^2\)[/tex] by expressing it as [tex]\((x^2)^2\)[/tex].
2. This relationship suggests a substitution might simplify the equation.
A useful substitution is:
- Let [tex]\(u = x^2\)[/tex].
With this substitution:
- [tex]\(x^4\)[/tex] can be written as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
Now substitute [tex]\(u = x^2\)[/tex] into the original equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex], so the equation becomes [tex]\(4u^2\)[/tex].
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
Therefore, 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], since it is in the standard form [tex]\(au^2 + bu + c = 0\)[/tex].
Thus, the substitution to rewrite the equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
Let's examine the terms in the equation:
1. The term [tex]\(x^4\)[/tex] can be related to [tex]\(x^2\)[/tex] by expressing it as [tex]\((x^2)^2\)[/tex].
2. This relationship suggests a substitution might simplify the equation.
A useful substitution is:
- Let [tex]\(u = x^2\)[/tex].
With this substitution:
- [tex]\(x^4\)[/tex] can be written as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].
Now substitute [tex]\(u = x^2\)[/tex] into the original equation:
- Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex], so the equation becomes [tex]\(4u^2\)[/tex].
- Replace [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex].
Therefore, 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], since it is in the standard form [tex]\(au^2 + bu + c = 0\)[/tex].
Thus, the substitution to rewrite the equation as a quadratic equation is [tex]\(u = x^2\)[/tex].
