Final answer:
To determine how long it takes for a rock to hit the ground, solve the quadratic equation representing its trajectory for the positive x-intercept, indicating the time when the height is zero.
Step-by-step explanation:
The question asks how long it takes for a rock to hit the ground, represented by the positive x-intercept of the given equation y = 9x^2 - 5x - 26, where y is the height of the rock and x is the time in seconds. To find when the rock hits the ground (y = 0), we need to solve the quadratic equation 9x^2 - 5x - 26 = 0.
We apply the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a), where a = 9, b = -5, and c = -26. The discriminant (√(b^2 - 4ac)) must be calculated first to find if real roots exist, which indicates potential times the rock could hit the ground. After computing the discriminant and applying the quadratic formula, we find two roots, but only the positive root is relevant because time cannot be negative.
The steps involved in the solution include computing the discriminant, applying the quadratic formula, and then interpreting the results to find the credible root for the time taken for the rock to reach the ground.