Solve the equation x^3 = 9x^2 + 22x.

Step-by-step solution

1. 1

   Set the equation to zero instead of dividing by $x$. Dividing both sides by $x$ would look attractive but would silently delete the root $x=0$. Move every term to the left:

   $x^3-9x^2-22x=0$

2. 2

   Factor out the common factor $x$. Every term contains at least one $x$:

   $x\left(x^2-9x-22\right)=0$

   This preserves $x=0$ as a genuine solution rather than losing it.

3. 3

   Factor the quadratic. Two numbers with product $-22$ and sum $-9$ are $-11$ and $+2$:

   $x^2-9x-22=(x-11)(x+2)\quad\Longrightarrow\quad x(x-11)(x+2)=0$

4. 4

   Apply the zero-product property to each factor.

   $x=0,\qquad x=11,\qquad x=-2$

5. 5

   Check all three in the original equation. At $x=11$: $11^3=1331$ and $9(121)+22(11)=1089+242=1331$ $\checkmark$. At $x=-2$: $-8$ and $9(4)-44=36-44=-8$ $\checkmark$. At $x=0$: $0=0$ $\checkmark$. A cubic has at most three roots, so this list is complete.