Patrick has a rectangular patio whose length is 5 m less than the diagonal and a width which is 7 m less than the diagonal. If the field of his patio is 195 m², what is the length of the diagonal?

Let x = the length of the diagonal. Thus, x - 5 = the length of the patio and x -7 = the width of the patio. Because the area is 195 m², and area is length times the width, the equation is (x - 5)(x - 7) = 195. Use the distributive property to multiply the binomials: x² -5x - 7x + 35 = 195. Combine such as terms: x² - 12x + 35 = 195. Subtract 195 from both sides: x² - 12x + 35 - 195= 195 - 195. Simplify: x2 - 12x - 160 = 0. Factor the result: (x - 20)(x + 8) = 0. Set every factor equal to 0 and solve: x - 20 = 0 or x + 8 = 0; x = 20 or x = -8. Reject the solution of -8 since a distance will not be negative. The length of the diagonal is 20 m.