The diagonal of the rectangle is 30 dm, and its area is 432 dm2. Find the sides of the rectangle.

Let’s immediately introduce two variables to solve the problem.

Let the length of the rectangle be m, the width of the rectangle n.

Let’s compose and solve a system of two equations with two unknowns:

m ^ 2 + n ^ 2 = 900;

m * n = 432;

Using the method of algebraic addition, we add the first equation and the second, multiplied by two:

m ^ 2 + n ^ 2 + 2 * m * n = 900 + 864;

(m + n) ^ 2 = 1764;

m + n = 42;

We get from here:

n = 42 – m;

Substitute the expression for the variable into the second equation of the system:

m * (42 – m) – 432 = 0;

42 * m – m ^ 2 – 432 = 0;

m ^ 2 – 42 * m + 432 = 0;

m ^ 2 – 2 * m * 21 + 441 – 9 = 0;

(m – 21) ^ 2 = 9;

m1 – 21 = -9;

m2 – 21 = 9;

m1 = 12;

m2 = 30.

The sides of the rectangle are 12 and 30 in.