Answer :
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we use substitution to simplify it.
Here's a step-by-step guide:
1. Identify the substitution: We will substitute [tex]\( u = x^2 \)[/tex]. This means that wherever we see [tex]\( x^2 \)[/tex], we will replace it with [tex]\( u \)[/tex].
2. Rewrite [tex]\( x^4 \)[/tex]: Since [tex]\( x^4 = (x^2)^2 \)[/tex], we can rewrite [tex]\( x^4 \)[/tex] as [tex]\( u^2 \)[/tex].
3. Substitute into the equation: Replace every occurrence of [tex]\( x^4 \)[/tex] with [tex]\( u^2 \)[/tex] and [tex]\( x^2 \)[/tex] with [tex]\( u \)[/tex]. The original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] becomes:
[tex]\[
4(u^2) - 21(u) + 20 = 0
\][/tex]
4. Recognize the quadratic form: Now, the equation [tex]\( 4u^2 - 21u + 20 = 0 \)[/tex] is a quadratic equation in terms of [tex]\( u \)[/tex].
By making the substitution [tex]\( u = x^2 \)[/tex], we have successfully rewritten the original equation as a quadratic equation.
Here's a step-by-step guide:
1. Identify the substitution: We will substitute [tex]\( u = x^2 \)[/tex]. This means that wherever we see [tex]\( x^2 \)[/tex], we will replace it with [tex]\( u \)[/tex].
2. Rewrite [tex]\( x^4 \)[/tex]: Since [tex]\( x^4 = (x^2)^2 \)[/tex], we can rewrite [tex]\( x^4 \)[/tex] as [tex]\( u^2 \)[/tex].
3. Substitute into the equation: Replace every occurrence of [tex]\( x^4 \)[/tex] with [tex]\( u^2 \)[/tex] and [tex]\( x^2 \)[/tex] with [tex]\( u \)[/tex]. The original equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] becomes:
[tex]\[
4(u^2) - 21(u) + 20 = 0
\][/tex]
4. Recognize the quadratic form: Now, the equation [tex]\( 4u^2 - 21u + 20 = 0 \)[/tex] is a quadratic equation in terms of [tex]\( u \)[/tex].
By making the substitution [tex]\( u = x^2 \)[/tex], we have successfully rewritten the original equation as a quadratic equation.
