What substitution should be used to rewrite [tex][tex]$16\left(x^3+1\right)^2 - 22\left(x^3+1\right) - 3 = 0$[/tex][/tex] as a quadratic equation?

A. [tex]u = x^3[/tex]
B. [tex]u = x^3 + 1[/tex]
C. [tex]u = (x^3 + 1)^2[/tex]
D. [tex]u = (x^3 + 1)^3[/tex]



Answer :

To solve the equation [tex]\(16(x^3 + 1)^2 - 22(x^3 + 1) - 3 = 0\)[/tex] and rewrite it as a quadratic equation, we can use an appropriate substitution.

Given the equation [tex]\(16(x^3 + 1)^2 - 22(x^3 + 1) - 3 = 0\)[/tex]:

1. We notice that the expression [tex]\(x^3 + 1\)[/tex] appears repeatedly. This suggests a substitution that can simplify it.

2. Let's denote [tex]\(u = x^3 + 1\)[/tex]. By making this substitution, the equation then simplifies into terms involving [tex]\(u\)[/tex] only.

3. Substituting [tex]\(u = x^3 + 1\)[/tex] into the equation, we have:
[tex]\[ 16(u)^2 - 22(u) - 3 = 0 \][/tex]

4. This is now a quadratic equation in terms of [tex]\(u\)[/tex], specifically:
[tex]\[ 16u^2 - 22u - 3 = 0 \][/tex]

Thus, the correct substitution to use is [tex]\(u = x^3 + 1\)[/tex].

Therefore, the answer is:
[tex]\[ \boxed{u = (x^3 + 1)} \][/tex]