What are the solutions to the equation $3(x-4)(x+5)=0$ ?
The equation to be solved is \(3(x-4)(x+5) = 0\). This is a product of factors equal to zero, where the factors are \(3\), \((x-4)\), and \((x+5)\). Since \(3\) is a constant and is not equal to zero, the equation can only be satisfied if one of the variable factors is zero. This is a direct application of the Zero Product Property, which states that for any product \(a \cdot b = 0\), at least one of the factors must be zero.
To simplify the equation \(3(x-4)(x+5) = 0\), we divide both sides by the constant factor \(3\):
\[\frac{3(x-4)(x+5)}{3} = \frac{0}{3} \implies (x-4)(x+5) = 0.\]
Now we apply the Zero Product Property, which states that \((x-4)(x+5) = 0\) if and only if \(x-4 = 0\) or \(x+5 = 0\).
We now solve the two resulting linear equations:
Thus, the solutions to the original equation are \(x = 4\) or \(x = -5\). Comparing this result with the given options, we see that it matches option b.
Understanding how to solve factored equations is essential for finding roots of functions, which can represent points where a graph crosses the x-axis. For example, if you are modeling the trajectory of a ball thrown in the air, the equation for its height might be factored to find the exact times it hits the ground. Solving \(3(x-4)(x+5)=0\) teaches you the fundamental logic used to determine these critical points, enabling you to solve more complex quadratic and higher-order equations in physics, engineering, and economics.