To rewrite the given equation [tex]\( 16(x^3 + 1)^2 - 22(x^3 + 1) - 3 = 0 \)[/tex] as a quadratic equation, we should use the substitution [tex]\( u = x^3 + 1 \)[/tex].
Here's a detailed, step-by-step explanation:
1. Identify the substitution variable:
Notice that the expression [tex]\( x^3 + 1 \)[/tex] appears in both the quadratic term and the linear term of the equation. To simplify the equation, we'll set [tex]\( u = x^3 + 1 \)[/tex].
2. Substitute [tex]\( u \)[/tex] into the equation:
By substituting [tex]\( u = x^3 + 1 \)[/tex], the equation becomes:
[tex]\[
16u^2 - 22u - 3 = 0
\][/tex]
3. Verify the new form:
Now the equation [tex]\( 16u^2 - 22u - 3 = 0 \)[/tex] is indeed a quadratic equation in terms of [tex]\( u \)[/tex].
4. Conclude the substitution:
Thus, the correct substitution to rewrite the original equation as a quadratic equation is [tex]\( u = x^3 + 1 \)[/tex].
So, the appropriate substitution is:
[tex]\[
u = x^3 + 1
\][/tex]
