What are the solutions to the equation $3(x-4)(x+5)=0$ ?
The given equation is \(3(x-4)(x+5) = 0\). This is a quadratic equation already presented in its factored form. To solve for \(x\), we need to determine the values that make the entire expression equal to zero. This is a classic application of the Zero Product Property.
The Zero Product Property states that if the product of multiple factors is equal to zero, then at least one of those factors must equal zero. In our equation \(3(x-4)(x+5) = 0\), we have three factors: \(3\), \((x-4)\), and \((x+5)\). Since the constant factor \(3\) is never zero, we can divide both sides of the equation by 3 to simplify it to \((x-4)(x+5) = 0\). Now, we set each variable factor equal to zero to find the possible values for \(x\).
We have two factors involving \(x\):
Thus, the solutions to the equation are \(x = 4\) and \(x = -5\). Comparing this result to the given options, we see that option b is the correct one.
Solving quadratic equations by factoring is a fundamental skill used in many practical areas such as physics and engineering. For instance, if you are modeling the trajectory of a ball thrown into the air, the height equation might take a form like \(h(t) = -5(t - 1)(t - 5)\), where \(t\) is time. To find when the ball hits the ground (\(h(t) = 0\)), you use the same factoring method: the ball is at ground level at \(t = 1\) second and \(t = 5\) seconds. This helps us predict movement and optimize designs in real-world scenarios.