What are the zeros of the equation shown?
$0=-3(x+2)(x-3)$
To find the zeros of the given equation \(0 = -3(x + 2)(x - 3)\), we need to determine the values of \(x\) that make the expression equal to zero. A polynomial expression equals zero when at least one of its factors is zero. This principle is known as the Zero Product Property. The equation involves three factors: \(-3\), \((x + 2)\), and \((x - 3)\). Since the constant factor \(-3\) is never zero, we only need to consider the variable-containing factors.
To identify the zeros, we set each factor containing the variable \(x\) to zero:
Solving the first equation:
\(x + 2 = 0 \implies x = -2\)
Solving the second equation:
\(x - 3 = 0 \implies x = 3\)
Thus, the zeros of the polynomial equation are \(x = -2\) and \(x = 3\). Expressed as coordinates \((x, 0)\), the zeros are \((-2, 0)\) and \((3, 0)\).
Comparing our findings to the given options:
Our calculated zeros, \((-2, 0)\) and \((3, 0)\), match option C. Therefore, the zeros are \((-2, 0)\) and \((3, 0)\).
Finding the zeros of a quadratic equation is a fundamental skill in algebra with many real-world applications. For example, if a company wants to model the profit of a product based on its price \(x\), they might use a quadratic equation like \(P(x) = -3(x + 2)(x - 3)\). The zeros represent the "break-even" points where the profit is exactly zero. Knowing these values helps the company identify the range of prices that will result in a profit. Analyzing the roots of such equations allows for better decision-making in economics, engineering, and physics.