$$ 1 \leftarrow \quad $$ Solve the equation. $$ (x-3)^{2 / 3}+(x-3)^{1 / 3}-6=0 $$ The solution set is $$ \{ $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
(x - 3)^{\frac{2}{3}} + (x - 3)^{\frac{1}{3}} - 6 = 0
$$

**Step 1: Substitute**

Let $$ y = (x - 3)^{\frac{1}{3}} $$. Then the equation becomes:

$$
y^2 + y - 6 = 0
$$

**Step 2: Solve the Quadratic Equation**

Solve for $$ y $$:

$$
y = \frac{-1 \pm \sqrt{1 + 24}}{2} = \frac{-1 \pm 5}{2}
$$

This gives two solutions:

$$
y = 2 \quad \text{and} \quad y = -3
$$

**Step 3: Back-Substitute to Find $$ x $$**

1. For $$ y = 2 $$:

$$
(x - 3)^{\frac{1}{3}} = 2 \implies x - 3 = 2^3 = 8 \implies x = 11
$$

2. For $$ y = -3 $$:

$$
(x - 3)^{\frac{1}{3}} = -3 \implies x - 3 = (-3)^3 = -27 \implies x = -24
$$

**Solution Set**

$$
\boxed{\{-24, \ 11\}}
$$
The solutions are $$ x = -24 $$ and $$ x = 11 $$.

\( 1 \leftarrow \quad \) Solve the equation. \( (x-3)^{2 / 3}+(x-3)^{1 / 3}-6=0 \) The solution set is \( \{ \)