Step-by-step explanation:
The sum of two positive numbers is 4 and the sum of their squares is 28. What are the two numbers?
Let’s write this as a pair of equations:
x+y=4
x2+y2=28
Rearrange the first equation:
y=4−x
and substitute into the second:
x2+(4−x)2=28
2x2−8x+16=28
x2−4x−6=0
Now by the quadratic formula:
x=4±42−4×1×(−6)√2×1=2±10−−√
Now 2−10−−√<0 so this solution is not of interest to us. But if x=2+10−−√ then y=2−10−−√ , so they fail to be both positive.
Hence the problem has no solution.
Update: Since I wrote this, I realised an even simpler way. Square the first equation:
x2+2xy+y2=16
and subtract the second equation from this, giving
2xy=−12
Since we require that x and y both be positive, this means 2xy is positive, hence we have a contradiction.
