If two positive numbers are such that the sum of their squar

If two positive numbers are such that the sum of their squares is 49 and the difference of their squares is 23. Calculate the smaller number.

Solution

We\'ll note the numbers as x and y and we\'ll decide, after calculus, which is the smaller number.

We\'ll impose the constraints given by enunciation and we\'ll get:

x^2 + y^2 = 49 (1)

x^2 - y^2 = 23 (2)

We\'ll add (1)+(2):

x^2 + y^2 + x^2 - y^2 = 49 + 23

We\'ll combine and eliminate like terms:

2x^2 = 72

We\'ll divide by 2:

x^2 = 36

Now, we can calculate y^2 from (1):

x^2 + y^2 = 49

36 + y^2 = 49

We\'ll subtract 36 both sides:

y^2 = 49-36

y^2 = 13

So, x = 6 and y = sqrt13 < 6

The smaller number is y = sqrt13.