Final answer:
To find (x² + y²), we can solve the two given equations to find the values of x and y. By substituting the value of y into the second equation, we can rearrange it into a quadratic equation in terms of x. Solving this quadratic equation gives us two values for x, which we can then substitute back to find the corresponding y values. Finally, substituting these values into (x² + y²) gives us the answer of 2672.142.
Step-by-step explanation:
To find the value of (x² + y²), we need to manipulate the equation given. From the first equation (xy = 6), we can solve for y = 6/x. Substituting this value of y into the second equation (x²y + xy² + x + y = 63), we get (x²(6/x) + x(6/x)² + x + 6/x = 63). Simplifying further, we have (6 + 6 + x + 6/x = 63). Combining like terms and multiplying through by x, we get (6x + 6x + x² + 6 = 63x). Rearranging to a quadratic equation form, we have (x² + 12x - 63x + 6 = 0). Simplifying this, we have (x² - 51x + 6 = 0). Now, we can solve this quadratic equation and find the values of x. Using the quadratic formula, we get two values for x: (x = 0.116) and (x = 50.884). Substituting these values back into the first equation (xy = 6), we can find the corresponding y values: (y = 51.724) and (y = 0.117). Finally, we can calculate the value of (x² + y²) using these values: (x² + y² = (0.116)² + (51.724)² = 2672.142). Therefore, the correct option is (d) (52).