Three squares have a total area of 35.25 𝑐𝑚2 . The larger square has twice the side-length of the middle-sized square. The smaller square has its side length exactly 0.5 cm smaller than the middle-sixed square. Find the side lengths of each of the three squares.

Let's assume the side length of the middle-sized square is x cm. According to the given information, the larger square has twice the side length of the middle-sized square. Therefore, the side length of the larger square is 2x cm. The smaller square has its side length exactly 0.5 cm smaller than the middle-sized square. So, the side length of the smaller square is (x - 0.5) cm. The area of a square is calculated by squaring its side length. Therefore, we can write the following equations based on the given information: x^2 + (2x)^2 + (x - 0.5)^2 = 35.25 Expanding the equation: x^2 + 4x^2 + x^2 - x + 0.25 = 35.25 Combining like terms: 6x^2 - x + 0.25 = 35.25 Subtracting 35 from both sides: 6x^2 - x - 35 = 0 Now, we have a quadratic equation. We can solve it using various methods, such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula: x = (-(-1) ± √((-1)^2 - 4 * 6 * -35)) / (2 * 6) Simplifying: x = (1 ± √(1 + 840)) / 12 x = (1 ± √841) / 12 x = (1 ± 29) / 12 Now, we have two possible values for x: 1. (1 + 29) / 12 = 30 / 12 = 2.5 2. (1 - 29) / 12 = -28 / 12 = -2.33 (discard because side lengths cannot be negative) Therefore, the side length of the middle-sized square is 2.5 cm. The side length of the larger square is twice the side length of the middle-sized square, so it is 2 * 2.5 = 5 cm. The side length of the smaller square is 0.5 cm smaller than the middle-sized square, so it is 2.5 - 0.5 = 2 cm. In summary, the side lengths of the three squares are as follows: Smaller square: 2 cm Middle-sized square: 2.5 cm Larger square: 5 cm