Solve x³ - 15x² + 75x - 125 = 0.

Step-by-step solution

1. 1

   Test the coefficients against the cube pattern. Expanding the template gives

   $(x-a)^3=x^3-3ax^2+3a^2x-a^3$

   Matching the $x^2$ coefficient: $-3a=-15$, so $a=5$. Now both remaining coefficients must agree, and they do: $3a^2=75$ ✓ and $-a^3=-125$ ✓. All three checks passing is what licenses the shortcut — had any one failed, the cubic would not be a perfect cube.

2. 2

   Rewrite the equation.

   $x^3-15x^2+75x-125=(x-5)^3=0$

3. 3

   Solve the cube. A cube equals zero only when its base does, so

   $x-5=0\quad\Rightarrow\quad x=5$

4. 4

   Record the multiplicity. The factor $(x-5)$ occurs three times, so $x=5$ is a triple root. A cubic has three roots counted with multiplicity, and here all three coincide — there are no other real roots and no complex ones.

5. 5

   Verify, and read the geometry. Substituting: $125-375+375-125=0$ ✓. Moreover $f'(x)=3x^2-30x+75=3(x-5)^2$ and $f''(x)=6x-30$, so $f'(5)=0$ and $f''(5)=0$ as well. A graph therefore flattens out completely at $x=5$ before crossing — the visual fingerprint of a triple root.