Answer :
To transform the equation
[tex]$$
4x^4 - 21x^2 + 20 = 0
$$[/tex]
into a quadratic form, we use a substitution that simplifies the quartic term. Notice that the equation involves [tex]$x^4$[/tex] and [tex]$x^2$[/tex], and [tex]$x^4$[/tex] can be written as [tex]$(x^2)^2$[/tex]. This suggests using the substitution
[tex]$$
u = x^2.
$$[/tex]
Now, substitute [tex]$u$[/tex] into the equation:
1. Replace [tex]$x^4$[/tex] with [tex]$u^2$[/tex] because [tex]$x^4 = (x^2)^2 = u^2$[/tex].
2. Replace [tex]$x^2$[/tex] with [tex]$u$[/tex].
The equation becomes
[tex]$$
4u^2 - 21u + 20 = 0.
$$[/tex]
This is now a standard quadratic equation in [tex]$u$[/tex]. Therefore, the correct substitution is
[tex]$$
u = x^2.
$$[/tex]
[tex]$$
4x^4 - 21x^2 + 20 = 0
$$[/tex]
into a quadratic form, we use a substitution that simplifies the quartic term. Notice that the equation involves [tex]$x^4$[/tex] and [tex]$x^2$[/tex], and [tex]$x^4$[/tex] can be written as [tex]$(x^2)^2$[/tex]. This suggests using the substitution
[tex]$$
u = x^2.
$$[/tex]
Now, substitute [tex]$u$[/tex] into the equation:
1. Replace [tex]$x^4$[/tex] with [tex]$u^2$[/tex] because [tex]$x^4 = (x^2)^2 = u^2$[/tex].
2. Replace [tex]$x^2$[/tex] with [tex]$u$[/tex].
The equation becomes
[tex]$$
4u^2 - 21u + 20 = 0.
$$[/tex]
This is now a standard quadratic equation in [tex]$u$[/tex]. Therefore, the correct substitution is
[tex]$$
u = x^2.
$$[/tex]
