In a right-angled triangle, the hypotenuse is 5, and one leg is 1 more than the other. Find the area of the triangle.

Let one of the legs be equal to x, then the second is equal to x + 1.
The sum of the squares of the legs is equal to the square of the hypotenuse, which means:
x ^ 2 + (x + 1) ^ 2 = 5 ^ 2;
x ^ 2 + x ^ 2 + 2x + 1 = 25;
2x ^ 2 + 2x-24 = 0;
x ^ 2 + x-12 = 0.
We find the discriminant:
D = b ^ 2-4ac = 1 ^ 2-4 * 1 * (- 12) = 1 + 48 = 49 = 7 ^ 2.
x1 = (- b-√D) / 2a = (- 1-7) / 2 = -4 – leg length cannot be less than zero;
x2 = (- b + √D) / 2a = (- 1 + 7) / 2 = 3.
Therefore, one of the legs is 3, the second is 3 + 1 = 4.
The area of the triangle is equal to half the product of the legs: S = 0.5 * 3 * 4 = 6.