When solving equations like (x minus 5) times (x plus 7) equals zero, we use the Zero Product Property. This property states that if a product of two factors equals zero, then at least one of the factors must be zero. For example, three times zero equals zero, and zero times seven equals zero.
In our equation (x minus 5) times (x plus 7) equals zero, we can identify two factors. Factor one is (x minus 5) and factor two is (x plus 7). According to the Zero Product Property, since their product is zero, either factor one equals zero OR factor two equals zero.
Let's solve the first factor: x minus 5 equals zero. To isolate x, we add 5 to both sides of the equation. This gives us x equals 5. On the number line, we can see that x equals 5 is the solution to this factor.
Now let's solve the second factor: x plus 7 equals zero. To isolate x, we subtract 7 from both sides of the equation. This gives us x equals negative 7. On our number line, we now have both solutions: x equals 5 from the first factor and x equals negative 7 from the second factor.
Our complete solution set is x equals 5 or x equals negative 7. Let's verify these solutions by substituting them back into the original equation. For x equals 5: (5 minus 5) times (5 plus 7) equals 0 times 12 equals 0. For x equals negative 7: (negative 7 minus 5) times (negative 7 plus 7) equals negative 12 times 0 equals 0. Both solutions check out, confirming that our answers are correct.