When we have a product of two factors equal to zero, the Zero Product Property tells us that at least one of those factors must be zero. For example, three times zero equals zero, and zero times five equals zero. This fundamental property is key to solving our equation.
Our equation is (x minus four) times (x plus six) equals zero. According to the Zero Product Property, either the first factor (x minus four) equals zero, or the second factor (x plus six) equals zero. This gives us two separate equations to solve.
Let's solve the first factor: x minus four equals zero. To isolate x, we add four to both sides of the equation. This gives us x minus four plus four equals zero plus four. Simplifying both sides, we get x equals four. So one solution is x equals four.
Now let's solve the second factor: x plus six equals zero. To isolate x, we subtract six from both sides. This gives us x plus six minus six equals zero minus six. Simplifying both sides, we get x equals negative six. So our second solution is x equals negative six.
We have found two solutions: x equals four and x equals negative six. Together, these form our complete solution set, which we can write as the set containing four and negative six. These are the only values of x that make our original equation true.
It's important to verify our solutions by substituting them back into the original equation. For x equals four: we substitute into (x minus four)(x plus six) to get (four minus four)(four plus six) equals (zero)(ten) equals zero. Check! For x equals negative six: we get (negative six minus four)(negative six plus six) equals (negative ten)(zero) equals zero. Check! Both solutions are correct.
We can also visualize our solutions graphically. The equation y equals (x minus four)(x plus six) represents a parabola. The solutions to our original equation are the x-intercepts of this parabola - the points where it crosses the x-axis. These occur at x equals four and x equals negative six, which matches our algebraic solutions exactly.