Perimeter of a right triangle = 30, hypotenuse = 13. find all sides

Let one leg of a right-angled triangle be x, and the other leg y. From the condition, the sum of the sides of a right-angled triangle, that is, its perimeter is 30. And the hypotenuse is 13. Let’s make an equation and express one side:
P = x + y + z;

x + y + 13 = 30;

y = 30 – 13 – x;

y = 17 – x;

Let’s use the Pythagorean theorem and compose the equation:
z² = x² + y²;

(x² + (17 – x) ²) = 13²;

(x² + (17 – x) ²) = 169;

x² + 289 – 34x + x² = 169;

2x² – 34x + 289 – 169 = 0;

2x² – 34x + 120 = 0;

Find the roots by solving the quadratic equation:

Let’s calculate the discriminant:

D = b² – 4ac = (- 34) ² – 4 * 2 * 120 = 1156 – 960 = 196;

D ›0 means:

x1 = (- b – √D) / 2a = (34 – √196) / 2 * 2 = (34 – 14) / 4 = 20/4 = 5;

x2 = (- b + √D) / 2a = (34 + √196) / 2 * 2 = (34 + 14) / 4 = 48/4 = 12;

Let’s find the second side:

y = 17 – x

If x1 = 5, then y1 = 17 – 5 = 12;

If x2 = 12, then y2 = 17 – 12 = 5;

Answer: the sides of the triangle are 5 and 12.