How many times does the graph of $4x = 32 - x^2$ cross the $x$-axis?
Answer (1)
Rewrite the equation as a quadratic equation: x 2 + 4 x − 32 = 0 .
Solve the quadratic equation by factoring: ( x + 8 ) ( x − 4 ) = 0 .
Find the solutions: x = − 8 and x = 4 .
The graph crosses the x-axis twice: 2 .
Explanation
Understanding the Problem We are given the equation 4 x = 32 − x 2 . We want to find how many times the graph of this equation crosses the x -axis. This is equivalent to finding the number of real solutions to the equation when y = 0 .
Rewriting the Equation First, we rewrite the equation in the standard quadratic form: x 2 + 4 x − 32 = 0 .
Solving the Quadratic Equation Now, we can solve the quadratic equation x 2 + 4 x − 32 = 0 . We can use the quadratic formula or try to factor the equation. Let's try factoring. We are looking for two numbers that multiply to -32 and add to 4. These numbers are 8 and -4. So, we can factor the equation as ( x + 8 ) ( x − 4 ) = 0 .
Finding the Solutions The solutions to the equation are x = − 8 and x = 4 . Since there are two distinct real solutions, the graph crosses the x -axis twice.
Using the Quadratic Formula Alternatively, we can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c , where a = 1 , b = 4 , and c = − 32 . The discriminant is D = b 2 − 4 a c = 4 2 − 4 ( 1 ) ( − 32 ) = 16 + 128 = 144 . Since the discriminant is positive, there are two distinct real solutions. The solutions are x = 2 − 4 ± 144 = 2 − 4 ± 12 . So, x = 2 − 4 + 12 = 2 8 = 4 and x = 2 − 4 − 12 = 2 − 16 = − 8 .
Final Answer Therefore, the graph of 4 x = 32 − x 2 crosses the x -axis two times.
Examples
Imagine you're designing a curved ramp for a skateboard park. The equation 4 x = 32 − x 2 can represent the shape of the ramp. The points where the ramp crosses the ground (x-axis) are crucial for the ramp's design and safety. By finding these points, you ensure a smooth transition for skateboarders, making the ramp functional and safe. This problem demonstrates how quadratic equations can be applied in real-world design scenarios to determine key intersection points.
