Input the sum of squares and the square of the sum to quickly find two numbers!
Let \(x\) and \(y\) be the two numbers. Suppose their sum of squares (\(S\)) and square of the sum (\(T\)) are given.
Solution:
1. Compute \(x + y\):
\( x + y = \sqrt{T} = \sqrt{441} = 21\)
2. Compute \(xy\):
\( xy = \frac{T - S}{2} = \frac{441 - 225}{2} = \frac{216}{2} = 108\)
3. Form the quadratic equation:
\( t^2 - 21t + 108 = 0\)
4. Solve for \(x\) and \(y\):
\( t = \frac{21 \pm \sqrt{21^2 - 4 \times 108}}{2} = \frac{21 \pm \sqrt{441 - 432}}{2} = \frac{21 \pm 3}{2}\)
\(x = 12\) and \(y = 9\)
Result: The two numbers are 12 and 9.
Solution:
1. Compute \(x + y\):
\( x + y = \sqrt{T} = \sqrt{196} = 14\)
2. Compute \(xy\):
\( xy = \frac{T - S}{2} = \frac{196 - 100}{2} = \frac{96}{2} = 48\)
3. Form the quadratic equation:
\( t^2 - 14t + 48 = 0\)
4. Solve for \(x\) and \(y\):
\( t = \frac{14 \pm \sqrt{14^2 - 4 \times 48}}{2} = \frac{14 \pm \sqrt{196 - 192}}{2} = \frac{14 \pm 2}{2}\)
\(x = 8\) and \(y = 6\).
Result: The two numbers are 8 and 6.
Solution:
1. Compute \(x + y\):
\( x + y = \sqrt{T} = \sqrt{625} = 25\)
2. Compute \(xy\):
\( xy = \frac{T - S}{2} = \frac{625 - 313}{2} = \frac{312}{2} = 156\)
3. Form the quadratic equation:
\( t^2 - 25t + 156 = 0\)
4. Solve for \(x\) and \(y\):
\( t = \frac{25 \pm \sqrt{25^2 - 4 \times 156}}{2} = \frac{25 \pm \sqrt{625 - 624}}{2} = \frac{25 \pm 1}{2}\)
\(x = 13\) and \(y = 12\).
Result: The two numbers are 13 and 12.