What are the zeros of the equation shown? $0=-3(x+2)(x-3)$
The problem asks to find the zeros of the quadratic equation \(0 = -3(x + 2)(x - 3)\). The zeros of a function are the \(x\)-values for which the function's value is equal to \(0\). In this case, we need to find the values of \(x\) that satisfy the given equation. We can identify the factors as \(-3\), \((x + 2)\), and \((x - 3)\). According to the zero product property, if the product of several factors is zero, then at least one of the factors must be zero. Therefore, we set each expression containing \(x\) equal to zero: \(x + 2 = 0\) and \(x - 3 = 0\). The constant factor \(-3\) cannot be zero.
Now, we solve the two linear equations for \(x\):
First, for \(x + 2 = 0\), subtracting 2 from both sides gives \(x = -2\).
Second, for \(x - 3 = 0\), adding 3 to both sides gives \(x = 3\).
Thus, the \(x\)-values that make the expression zero are \(x = -2\) and \(x = 3\). Consequently, the zeros are the coordinate pairs \((-2, 0)\) and \((3, 0)\). Comparing these results to the given options, we find that this matches option C.
Finding the zeros of a function is crucial in real-world applications such as physics and engineering, where you need to determine when an object, like a projectile, hits the ground. For instance, if the height of an object is modeled by a quadratic equation, its zeros represent the times when the object's height is zero, helping you calculate exactly when it reaches its starting point or strikes the earth. Understanding how to solve these equations is a foundational skill for analyzing motion and designing systems.