Solve (x-4)² + x(x-12) = 16: Finding Parameter x

This is a quadratic equation (highest power is $x^2$), and quadratic equations can have up to two solutions. When you factor to get $2x(x-10) = 0$, either factor can equal zero!

Remember that $(x-4)^2 = (x-4)(x-4)$. Use FOIL: First + Outer + Inner + Last = $x^2 - 4x - 4x + 16 = x^2 - 8x + 16$

If factoring doesn't work easily, you can always use the quadratic formula. But in this case, factoring out the common factor $2x$ makes it much simpler!

Substitute each solution back into the original equation. For $x = 0$: $(0-4)^2 + 0(0-12) = 16 + 0 = 16$ ✓
For $x = 10$: $(10-4)^2 + 10(10-12) = 36 + (-20) = 16$ ✓

This uses the Zero Product Property: if two factors multiply to give zero, then at least one factor must equal zero. So from $2x(x-10) = 0$, either $2x = 0$ or $x-10 = 0$!