Answer :
To solve the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] by rewriting it as a quadratic equation, we can use a substitution method. Here are the step-by-step instructions for this process:
1. Consider a Substitution: We want to simplify the equation by substituting a variable for part of the expression. Let's set [tex]\( u = x^2 \)[/tex]. This means that wherever we see [tex]\( x^2 \)[/tex] in the equation, we will replace it with [tex]\( u \)[/tex].
2. Express Higher Powers in Terms of [tex]\( u \)[/tex]: Since [tex]\( u = x^2 \)[/tex], then [tex]\( x^4 = (x^2)^2 = u^2 \)[/tex]. This allows us to express [tex]\( 4x^4 \)[/tex] as [tex]\( 4u^2 \)[/tex].
3. Rewrite the Original Equation: Substitute the expressions in terms of [tex]\( u \)[/tex] into the original equation:
- Original: [tex]\( 4x^4 - 21x^2 + 20 = 0 \)[/tex]
- Substitute: [tex]\( 4u^2 - 21u + 20 = 0 \)[/tex]
4. Recognize the Quadratic Form: Now, the equation [tex]\( 4u^2 - 21u + 20 = 0 \)[/tex] is a quadratic equation in the variable [tex]\( u \)[/tex].
By using the substitution [tex]\( u = x^2 \)[/tex], we successfully transformed the original equation into a quadratic form. This substitution simplifies solving the equation by an easier method, such as solving a standard quadratic equation.
1. Consider a Substitution: We want to simplify the equation by substituting a variable for part of the expression. Let's set [tex]\( u = x^2 \)[/tex]. This means that wherever we see [tex]\( x^2 \)[/tex] in the equation, we will replace it with [tex]\( u \)[/tex].
2. Express Higher Powers in Terms of [tex]\( u \)[/tex]: Since [tex]\( u = x^2 \)[/tex], then [tex]\( x^4 = (x^2)^2 = u^2 \)[/tex]. This allows us to express [tex]\( 4x^4 \)[/tex] as [tex]\( 4u^2 \)[/tex].
3. Rewrite the Original Equation: Substitute the expressions in terms of [tex]\( u \)[/tex] into the original equation:
- Original: [tex]\( 4x^4 - 21x^2 + 20 = 0 \)[/tex]
- Substitute: [tex]\( 4u^2 - 21u + 20 = 0 \)[/tex]
4. Recognize the Quadratic Form: Now, the equation [tex]\( 4u^2 - 21u + 20 = 0 \)[/tex] is a quadratic equation in the variable [tex]\( u \)[/tex].
By using the substitution [tex]\( u = x^2 \)[/tex], we successfully transformed the original equation into a quadratic form. This substitution simplifies solving the equation by an easier method, such as solving a standard quadratic equation.
