Question

Solve the cube root equation x3−12x2+16x+48−−−−−−−−−−−−−−−−−√3=x−4.

Answer 1

To solve the equation

\[
\sqrt[3]{x^3 - 12x^2 + 16x + 48} = x - 4,
\]

we begin by cubing both sides to eliminate the cube root:

\[
x^3 - 12x^2 + 16x + 48 = (x - 4)^3.
\]

Next, we expand the right-hand side:

\[
(x - 4)^3 = x^3 - 12x^2 + 48x - 64.
\]

Now we can rewrite the equation as:

\[
x^3 - 12x^2 + 16x + 48 = x^3 - 12x^2 + 48x - 64.
\]

Next, we subtract \(x^3 - 12x^2\) from both sides:

\[
16x + 48 = 48x - 64.
\]

Now, we can rearrange the equation:

\[
16x + 48 - 48x + 64 = 0,
\]
which simplifies to

\[
-32x + 112 = 0.
\]

Next, we solve for \(x\):

\[
-32x = -112 \Rightarrow x = \frac{112}{32} = \frac{7}{2} = 3.5.
\]

Now we need to verify if this solution satisfies the original equation. We will substitute \(x = 3.5\) back into the left-hand side:

First we calculate:

\[
x^3 - 12x^2 + 16x + 48 = (3.5)^3 - 12(3.5)^2 + 16(3.5) + 48.
\]

Calculating each term:

\[
(3.5)^3 = 42.875,
\]

\[
12(3.5)^2 = 12 \times 12.25 = 147,
\]

\[
16(3.5) = 56.
\]

Adding these gives:

\[
42.875 - 147 + 56 + 48.
\]

Calculating this step by step:

- First, \(42.875 + 56 = 98.875\).
- Then, \(98.875 - 147 = -48.125\).
- Finally, \(-48.125 + 48 = -0.125\).

Now we calculate the right-hand side \(x - 4\):

\[
3.5 - 4 = -0.5.
\]

Lastly, we can check if the original equation holds true:

Calculating the cube root on the left-hand side:

\[
\sqrt[3]{-0.125} = -0.5.
\]

As both left-hand side and right-hand side equal \(-0.5\), the solution satisfies the equation.

Thus, the solution to the equation is

\[
\boxed{3.5}.
\]