The area of the rectangle is calculated by the formula

A = l * w

'l' = length of the rectangle

'w' = width of the rectangle

'A' = area of the rectangle

Let

l1 and w1 by the length and the width of the first rectangle

l1 and w1 by the length and the width of the second rectangle

l1 * w1 = l2 * w2 = 630 the two rectangles have an area of 630 cm2 each

|l1 - l2| = 5 difference in the lengths of the rectangles is 5 cm

|w1 - w2| = 3 difference in the width of the rectangles is 3 cm

We need to find l1, w1, l2, w2 positive numbers.

Without loss of generality we can assume that l1 >= l2 then |l1 - l2| = l1 - l2 (by the definition for absolute value)

Also if l1 > = l2 then w1 <= w2 is a must (we prove this using the fact that l1 * w1 = l2 * w2 and l1 > = l2 if w1 > w2 then l1 * w1 > l2 * w2 and we can not have that (contradiction))

Now we have the following equations

l1 * w1 = 630 = l2 * w2

l1 - l2 = 5

|w1 - w2| = 3

Since w1 < = w2 that is |w1 - w2| = - (w1 - w2) = w2 - w1 (by the definition for absolute value) we have the following equations

l1 - l2 = 5 => l1 = 5 + l2

w2 - w1 = 3 => w1 = w2 - 3

l1 * w1 = 630 => (5 + l2) (w2 - 3) =630

l2 * w2 = 630

Lets pay attention to these 2 equations

l2 * w2 = 630

(5 + l2) ( w2 - 3) = 630

the second equation is equivalent to

5 * w2 - 5 *3 + l2 * w2 - 3* l2 = 630

5 * w2 - 15 + l2 * w2 - 3* l2 = 630

5 * w2 - 15 + 630 - 3* l2 = 630

5 * w2 - 3* l2 = 15

by replacing w2 with w2 = 630 / l2 in the last equation we get

5 * 630 / l2 - 3 * l2 = 15 multiply both sides by (- l2)

3* l2 ^ 2 - 5* 630 = - 15 * l2 divide both sides by 3

l2 ^ 2 - 1050 = - 5* l2

l2 + 5 * l2 - 1050 = 0 (quadratic equation)

l2 = (-5 + ((5^2 - 4* (- 1050))^ (1/2)) / 2

l2 = (-5 + 65 ) / 2

l2 = 30 cm - one solution

l2 = (-5 - ((5^2 - 4* (- 1050))^ (1/2)) / 2

l2 = (-5 - 65 ) / 2

l2 = - 35 - is not a solution since l2 needs to be positive

Now we know l2 = 30 cm. Lets go back to finding l1, w1, w2

w2 = 630 / l2 = 630 / 30 = 21 cm

l1 = 5 + l2 = % + 30 = 35 cm > l2

w1 = w2 - 3 = 21 - 3 = 18 cm < w2

One rectangle has length 35 cm and width 18 cm and the other rectangle has length 30 cm and width 21 cm.