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What substitution should be used to rewrite [tex][tex]$4x^4 - 21x^2 + 20 = 0$[/tex][/tex] as a quadratic equation?

A. [tex]u = x^2[/tex]
B. [tex]u = 2x^2[/tex]
C. [tex]u = x^4[/tex]
D. [tex]u = 4x^4[/tex]

Sagot :

GinnyAnswer
To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we can use a suitable substitution. Let's analyze the given equation step by step.

1. Notice the term [tex]\(4x^4\)[/tex] and [tex]\( -21x^2\)[/tex] in the equation. With the goal of rewriting it into a quadratic form, we can make a substitution.

2. One effective substitution for simplifying such expressions is to let a new variable [tex]\(u\)[/tex] represent [tex]\(x^2\)[/tex]. By doing this:
[tex]\[ u = x^2 \][/tex]

3. Plugging [tex]\(u = x^2\)[/tex] into the original equation, we obtain:
[tex]\[ 4(x^2)^2 - 21(x^2) + 20 = 0 \][/tex]

4. Simplify this substitution:
[tex]\[ 4u^2 - 21u + 20 = 0 \][/tex]

Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in [tex]\(u\)[/tex].

Thus, the correct substitution you should use is:
[tex]\[ u = x^2 \][/tex]
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