The width of the rectangle is less than the length by 4 cm, and its area is 77 cm2. Find the length and width of this rectangle.

We denote by x the width (in cm), and by y the length (also in cm) of the rectangle.
Then, according to the first condition of the task, x = y – 4.
As you know, the area (S) of a rectangle is calculated by the formula S = a * b, where a is the width, b is the length of the rectangle. According to the second condition, x * y = 77. Substitute instead of x the expression y – 4. Then, we get: (y – 4) * y = 77 or, expanding the parentheses, y * y – 4 * y = 77, whence y ^ 2 – 4 * y – 77 = 0.
The last quadratic equation has two different roots: y1 = 11 and y2 = –7 (subsidiary root). So, y = 11. Now we calculate x = y – 4 = 11 – 4 = 7.
Thus, the length of the rectangle is 11 cm and the width is 7 cm.
Answer: 11 cm and 7 cm.