Verified Answer :

  • Recognize that the expression equals zero when one of its factors is zero.
  • Set each linear factor, \((x + 2)\) and \((x - 3)\), to zero and solve for \(x\).
  • Identify the roots as \(x = -2\) and \(x = 3\), resulting in x-intercepts \((-2, 0)\) and \((3, 0)\).
  • Conclude that the correct option is \(\boxed{C}\).

Explanation

  1. Analyzing the Equation and Zero Product Property

To find the zeros of the equation \(0 = -3(x + 2)(x - 3)\), we identify the points where the function equals zero. The equation is already in factored form, which is a product of three factors: \(-3\), \((x + 2)\), and \((x - 3)\). The Zero Product Property states that a product of factors is zero if and only if at least one of the individual factors is zero.

  1. Solving for Zeros

We examine each factor individually to see if it can be zero:

  • The constant factor \(-3\) can never be zero.
  • For the factor \((x + 2)\) to be zero, we solve:

\[x + 2 = 0 \implies x = -2\]

  • For the factor \((x - 3)\) to be zero, we solve:

\[x - 3 = 0 \implies x = 3\]
Thus, the values of \(x\) that make the entire expression equal to zero are \(x = -2\) and \(x = 3\).

  1. Identifying the Zeros as Coordinates

An x-intercept of a function occurs when the value of the function (the y-coordinate) is \(0\). Since we found that the function equals zero at \(x = -2\) and \(x = 3\), the corresponding x-intercepts are \((-2, 0)\) and \((3, 0)\). Comparing this to the provided options, it corresponds to Option C.

Examples

In physics, the path of a projectile can be modeled by a quadratic equation, where the zeros represent the ground level. If the height \(h\) of an object is given by \(h(t) = -3(t + 2)(t - 3)\), the zeros help determine when the object is at ground level (ignoring negative time). Finding these points allows engineers and scientists to predict the exact time a projectile will land or be launched, helping in trajectory analysis and safety calculations.

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