We are solving the equation (x minus 3) times (x plus 5) equals zero. This is based on the Zero Product Property, which states that if a product of two factors is zero, then at least one of the factors must be zero. For example, 3 times 0 equals 0, and 0 times 7 equals 0. This property is fundamental to solving factored quadratic equations.
In the equation (x minus 3) times (x plus 5) equals zero, we can identify two factors. Factor 1 is (x minus 3) and Factor 2 is (x plus 5). According to the Zero Product Property, since their product is zero, at least one of these factors must equal zero. This means either (x minus 3) equals zero or (x plus 5) equals zero.
Since the product of (x minus 3) and (x plus 5) equals zero, we can set up two separate equations based on the Zero Product Property. Either the first factor (x minus 3) equals zero, or the second factor (x plus 5) equals zero. This gives us Equation 1: x minus 3 equals zero, and Equation 2: x plus 5 equals zero.
Let's solve the first equation: x minus 3 equals zero. To isolate x, we add 3 to both sides of the equation. This gives us x minus 3 plus 3 equals 0 plus 3, which simplifies to x equals 3. We can verify this solution by substituting back into the factor: (3 minus 3) equals 0, which checks out. On the number line, we mark x equals 3.
Now let's solve the second equation: x plus 5 equals zero. To isolate x, we subtract 5 from both sides. This gives us x plus 5 minus 5 equals 0 minus 5, which simplifies to x equals negative 5. We verify this by substituting back: (negative 5 plus 5) equals 0, which is correct. On our number line, we now have both solutions: x equals 3 and x equals negative 5.
Let's verify both solutions in the original equation. For x equals 3: (3 minus 3) times (3 plus 5) equals (0) times (8) equals 0, which checks out. For x equals negative 5: (negative 5 minus 3) times (negative 5 plus 5) equals (negative 8) times (0) equals 0, which also checks out. Therefore, the complete solution set is x equals 3 and x equals negative 5. On the coordinate plane, these are the points where the parabola y equals (x minus 3) times (x plus 5) crosses the x-axis.