Final answer:
To show that the x-coordinates of the intersection points A and B of the line y = 2x + p and the circle x² + y² = 73 satisfy the equation 5x² + 4px - 73 = 0, we substitute the expression for y from the linear equation into the circle's equation. After expanding and rearranging terms, assuming p² - 73 = 0, we arrive at the quadratic equation. It should be noted that there are no integer values for p which make p² equal to 73, indicating a potential issue with the question's parameters.
Step-by-step explanation:
The intersection of the line y = 2x + p and the circle x² + y² = 73 can be found by substituting the linear equation into the circle's equation. This substitution results in x² + (2x + p)² = 73. Expanding and simplifying this equation will give us the quadratic equation that the x-coordinates of points A and B satisfy.
Expanding the binomial, we get x² + (4x² + 4px + p²) = 73. Combining like terms results in 5x² + 4px + p² = 73. To show that this equation takes the form of 5x² + 4px - 73 = 0, we simply set the equation equal to zero and subtract 73 to the other side, assuming that p² - 73 equals zero. This means that p must be an integer whose square is 73 to satisfy the equation.
However, because there are no integer values for p such that p² = 73, it's likely that there is a typo in the initial question and p might have been defined differently since 73 is not a perfect square. Therefore, we must assume that the equation given by the question itself already includes whatever the value of p² that makes this equation true (e.g., through factoring in other constraints or typos in the presentation of the problem)