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

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



Answer :

GinnyAnswer
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]

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