Input the sum and sum of squares, and instantly find the two numbers.
Given:
Steps:
Solution:
1. Compute \( xy \):
\( xy = \frac{20^2 - 208}{2} = \frac{400 - 208}{2} = \frac{192}{2} = 96 \)
2. Solve the quadratic equation:
\( t^2 - 20t + 96 = 0 \)
3. Calculate the discriminant:
\( \sqrt{20^2 - 4 \cdot 96} = \sqrt{400 - 384} = \sqrt{16} = 4 \)
4. Find the roots:
\( x, y = \frac{20 \pm 4}{2} = 12 \text{ and } 8\)
Result: The numbers are 12 and 8.
Solution:
1. Compute \( xy \):
\( xy = \frac{25^2 - 325}{2} = \frac{625 - 325}{2} = \frac{300}{2} = 150 \)
2. Solve the quadratic equation:
\( t^2 - 25t + 150 = 0 \)
3. Calculate the discriminant:
\( \sqrt{25^2 - 4 \cdot 150} = \sqrt{625 - 600} = \sqrt{25} = 5 \)
4. Find the roots:
\( x, y = \frac{25 \pm 5}{2} = 15 \text{ and } 10 \)
Result: The numbers are 15 and 10.
Solution:
1. Compute \( xy \):
\( xy = \frac{34^2 - 610}{2} = \frac{1156 - 610}{2} = \frac{546}{2} = 273 \)
2. Solve the quadratic equation:
\( t^2 - 34t + 273 = 0 \)
3. Calculate the discriminant:
\( \sqrt{34^2 - 4 \cdot 273} = \sqrt{1156 - 1092} = \sqrt{64} = 8 \)
4. Find the roots:
\( x, y = \frac{34 \pm 8}{2} = 21 \text{ and } 13 \)
Result: The numbers are 21 and 13.