Input the product and sum of squares, and instantly find the two numbers.
Given:
Steps:
Solution:
1. Calculate \( x + y \):
\( x + y = \sqrt{404 + 2 \cdot 40} = \sqrt{484} = 22 \)
2. Form the quadratic equation:
\( t^2 - 22t + 40 = 0 \)
3. Calculate the discriminant:
\( \sqrt{22^2 - 4 \cdot 40} = \sqrt{484 - 160} = \sqrt{324} = 18 \)
4. Solve for \( t \):
\( t = \frac{22 \pm 18}{2} \implies t = 20 \text{ or } 2 \)
Result: The numbers are 20 and 2.
Solution:
1. Calculate \( x + y \):
\( x + y = \sqrt{289 + 2 \cdot 120} = \sqrt{529} = 23 \)
2. Form the quadratic equation:
\( t^2 - 23t + 120 = 0 \)
3. Calculate the discriminant:
\( \sqrt{23^2 - 4 \cdot 120} = \sqrt{529 - 480} = \sqrt{49} = 7 \)
4. Solve for \( t \):
\( t = \frac{23 \pm 7}{2} \implies t = 15 \text{ or } 8 \)
Result: The numbers are 15 and 8.
Solution:
1. Calculate \( x + y \):
\( x + y = \sqrt{225 + 2 \cdot 108} = \sqrt{441} = 21 \)
2. Form the quadratic equation:
\( t^2 - 21t + 108 = 0 \)
3. Calculate the discriminant:
\( \sqrt{21^2 - 4 \cdot 108} = \sqrt{441 - 432} = \sqrt{9} = 3 \)
4. Solve for \( t \):
\( t = \frac{21 \pm 3}{2} \implies t = 12 \text{ or } 9 \)
Result: The numbers are 12 and 9.