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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 should use the following substitution:

1. Let's use [tex]\(u = x^2\)[/tex]. This substitution will help us reduce the polynomial with degree 4 to a quadratic polynomial.

2. First, observe how the terms of the original equation transform:
- [tex]\(4x^4\)[/tex]: Since [tex]\(u = x^2\)[/tex], then [tex]\(x^4 = (x^2)^2 = u^2\)[/tex]. Thus, [tex]\(4x^4\)[/tex] becomes [tex]\(4u^2\)[/tex].
- [tex]\(-21x^2\)[/tex]: With [tex]\(u = x^2\)[/tex], [tex]\(-21x^2\)[/tex] becomes [tex]\(-21u\)[/tex].
- The constant term [tex]\(20\)[/tex] remains the same.

3. Substituting these into the original equation, we get:
[tex]\[ 4u^2 - 21u + 20 = 0 \][/tex]

Therefore, the correct substitution to rewrite [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation is:
[tex]\[ u = x^2 \][/tex]

So, the proper substitution is:
[tex]\[ \boxed{u = x^2} \][/tex]
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