Solve the equation 8x^2 = 16x.

Step-by-step solution

1. 1

   Resist the temptation to divide both sides by $x$. Dividing gives $8x=16$ and hence $x=2$ — but it loses the solution $x=0$, because dividing by $x$ silently assumes $x\neq0$. Division by a quantity that might be zero is never a safe move; factoring is.

2. 2

   Move everything to one side. Subtract $16x$ from both sides so that the right-hand side is zero — the form the zero-product property requires:

   $8x^{2}-16x=0$

3. 3

   Factor out the greatest common factor. The terms share $8$ and one $x$, so the GCF is $8x$:

   $8x(x-2)=0$

   Check by expanding: $8x\cdot x-8x\cdot2=8x^{2}-16x$ ✓.

4. 4

   Apply the zero-product property. A product is zero only if a factor is zero:

   $8x=0\quad\Longrightarrow\quad x=0,\qquad x-2=0\quad\Longrightarrow\quad x=2$

   The constant $8$ can never be zero, so it contributes no root — only the $x$ and the $(x-2)$ do.

5. 5

   Verify both roots in the original equation. At $x=0$: left $8(0)=0$, right $16(0)=0$ ✓. At $x=2$: left $8(4)=32$, right $16(2)=32$ ✓. Both check out, confirming that a quadratic here really does have its full complement of two roots.

6. 6

   Note the general lesson. Any quadratic with no constant term, $ax^{2}+bx=0$, factors as $x(ax+b)=0$ and therefore always has $x=0$ as one root, with the other at $x=-b/a$. Here $-(-16)/8=2$ ✓.