A rectangle with an area of 108 cm2. The diagonal is 15 cm. Find the sides of the rectangle.

Let one side of the rectangle be x, then the other side is 108 / x. The sides of the rectangle and its diagonal form a right-angled triangle to which the Pythagorean theorem can be applied (the diagonal is the hypotenuse, the sides are the legs).

x ^ 2 + (108 / x) ^ 2 = 15 ^ 2;

x ^ 2 + 11664 / (x ^ 2) – 225 = 0;

(x ^ 4 + 11664 – 225x ^ 2) / x ^ 2 = 0;

a fraction is zero when the numerator is zero and the denominator is nonzero;

x ^ 4 + 11664 – 225x ^ 2 = 0;

x ^ 4 – 225x ^ 2 + 11664 = 0;

we introduce a new variable y = x ^ 2;

y ^ 2 – 225y +11664 = 0;

D = b ^ 2 – 4ac;

D = (-225) ^ 2 – 4 * 1 * 11664 = 50625 – 46656 = 3969; √D = 63;

x = (-b ± √D) / (2a);

y1 = (225 + 63) / 2 = 288/2 = 144;

y2 = (225 – 63) / 2 = 162/2 = 81;

x ^ 2 = 144;

x = ± 12, side length cannot be expressed as a negative number, x = 12;

x ^ 2 = 81;

x = ± 9, side length cannot be expressed as a negative number, x = 9.

Answer. 12 cm, 9 cm.