Question

solving quadratic equations using common factors quick check as isla prepares to set off fireworks, she wants to ensure the spectators are a safe distance away. when shot in the air, one firework will follow the path $h(x) = -x^2 + 15x$, where $h(x)$ is the height of the firework, and $x$ is the horizontal distance it travels in feet. to determine how far the firework will travel before it reaches the ground, isla sets the height equal to zero, and solves the following quadratic equation: $0 = -x^2 + 15x$. how far will the firework travel before it reaches the ground? (1 point) 7.5 feet 15 feet -15 feet 30 feet

Explanation:

Step1: Factor the equation

Given the equation \(0 = -x^2 + 15x\), we can factor out \(-x\) (or \(x\)) from the right - hand side. Factoring out \(x\) gives us \(0=x(-x + 15)\) or \(0=-x(x - 15)\).

Step2: Apply the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
For the equation \(0=-x(x - 15)\), we have two cases:
Case 1: \(-x=0\), which gives \(x = 0\). This represents the initial position of the firework (when it is shot from the ground).
Case 2: \(x - 15=0\), which gives \(x=15\). This represents the position of the firework when it lands back on the ground. Since distance cannot be negative in this context, we discard the solution \(x = 0\) (as it is the starting point) and take \(x = 15\).

Answer:

15 feet