If the number y is 6 less than the number x and the sum of the squares of these two numbers is 68, find the product of two numbers.

Step 1: Understand the problem The problem gives us two pieces of information. The first is that y is 6 less than x, which can be written as y = x - 6. The second is that the sum of the squares of x and y is 68, which can be written as x² + y² = 68.

Step 2: Substitute y in the second equation We can substitute y in the second equation with x - 6. This gives us x² + (x - 6)² = 68.

Step 3: Simplify the equation The equation simplifies to x² + x² - 12x + 36 = 68.

Step 4: Combine like terms This further simplifies to 2x² - 12x - 32 = 0.

Step 5: Solve for x We can solve for x by factoring, completing the square, or using the quadratic formula. In this case, the quadratic formula is the easiest method. The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / 2a. Substituting the coefficients gives x = [12 ± sqrt((12)² - 42(-32))] / 2*2. Simplifying this gives x = 6 ± sqrt(36 + 256) / 4 = 6 ± sqrt(292) / 4. Therefore, the solutions for x are x = 6 + sqrt(292) / 4 and x = 6 - sqrt(292) / 4.

Step 6: Find y for each solution of x Substitute each solution for x into the equation y = x - 6 to find the corresponding y values.

Step 7: Find the product of x and y for each pair The product of x and y is xy. Substitute the pairs of x and y values into this equation to find the product for each pair.