Solve the quadratic equation.
$3 x=0.5 x^2$
The equation to solve is \(3x = 0.5x^2\). This is a quadratic equation which can be written in the standard form \(ax^2 + bx + c = 0\) by moving all terms to one side of the equation. Subtracting \(3x\) from both sides gives \(0.5x^2 - 3x = 0\).
To solve the equation \(0.5x^2 - 3x = 0\), we look for common factors. We can see that both terms share a factor of \(x\). Factoring out \(x\) gives \(x(0.5x - 3) = 0\). According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible cases: \(x = 0\) or \(0.5x - 3 = 0\).
The first case, \(x = 0\), is already solved. For the second case, we solve the linear equation \(0.5x - 3 = 0\). Adding 3 to both sides gives \(0.5x = 3\). Dividing both sides by 0.5, we get \(x = 6\). Therefore, the solutions to the equation are \(x = 0\) and \(x = 6\). Comparing this result with the provided options, we see that it matches option d.
Quadratic equations are fundamental in modeling real-world physical systems. For example, if you are throwing a ball, the height of the ball over time is described by a quadratic equation. Solving such equations allows us to determine when the ball hits the ground (height = 0) or at what time it reaches its maximum height. In this problem, solving \(3x = 0.5x^2\) is equivalent to finding the moments in time where a specific physical quantity governed by this equation is zero, which is a common task in physics and engineering.