Final answer:
The center of the circle represented by the equation x² + y² - 3x - 14 = 0 is at the coordinates (1.5, 0) after completing the square for the x-terms.
Step-by-step explanation:
To find the center of the circle described by the equation x² + y² - 3x - 14 = 0, we should complete the square for both x and y. Currently, there are no y-term coefficients aside from the squared term, so we only need to focus on the x-terms.
First, we rewrite the equation by isolating the linear x-term:
Next, we complete the square for x by adding (3/2)² to both sides of the equation:
- x² - 3x + (3/2)² + y² = 14 + (3/2)²
- (x - 3/2)² + y² = 14 + 9/4
- (x - 3/2)² + y² = 14 + 2.25
- (x - 3/2)² + y² = 16.25
The equation now represents a circle with center at (3/2, 0) and a radius of √16.25. However, we need to remember that this is not the original equation; to get back to the original circle, we will need to move the center back along the y-axis to where there are no y-terms in the original equation's left side. This indicates the y-coordinate of the center is 0.
Therefore, the center of the circle is at (3/2, 0), which after simplifying the fraction for the x-coordinate is (1.5, 0).