Find the sides of a rectangle if its area is 12 dm2 and the perimeter is 14 dm.

1. Let’s denote the sides of the rectangle by X dm and Y dm.
2. Then its area S = X * Y = 12 dm2.
3. The perimeter of the rectangle P is P = X + X + Y + Y = 2 * X + 2 * Y = 14 dm.
4. From the second equation Y = 7 – X.
5. Substitute the expression for Y into the first equation. We get a quadratic equation:
X ^ 2 – 7 * X + 12 = 0.
6. The discriminant of the equation D2 = 49 – 48 = 1. The roots of the equation are X = 4 and X = 3. Accordingly,
Y = 7 – 4 = 3 and Y = 7 – 3 = 4. That is, the sides of the rectangle are 3 dm and 4 dm.
Answer: the sides of the rectangle are 3 in. And 4 in.