Input the sum and sum of cubes to quickly find the two numbers.
Given:
Steps:
Solution:
1. Calculate \( xy \):
\( xy = \frac{11^3 - 737}{3 \cdot 11} = 18 \)
2. Form the quadratic equation:
\( t^2 - 11t + 18 = 0 \)
3. Calculate the discriminant:
\( \sqrt{11^2 - 4 \cdot 18} = \sqrt{121 - 72} = \sqrt{49} = 7 \)
4. Solve for \( t \):
\( t = \frac{11 \pm 7}{2} \implies t = 9 \text{ or } 2 \)
Result: The numbers are 9 and 2.
Solution:
1. Calculate \( xy \):
\( xy = \frac{9^3 - 189}{3 \cdot 9} = 20 \)
2. Form the quadratic equation:
\( t^2 - 9t + 20 = 0 \)
3. Calculate the discriminant:
\( \sqrt{9^2 - 4 \cdot 20} = \sqrt{81 - 80} = \sqrt{1} = 1 \)
4. Solve for \( t \):
\( t = \frac{9 \pm 1}{2} \implies t = 5 \text{ or } 4 \)
Result: The numbers are 5 and 4.
Solution:
1. Calculate \( xy \):
\( xy = \frac{8^3 - 152}{3 \cdot 8} = 15 \)
2. Form the quadratic equation:
\( t^2 - 8t + 15 = 0 \)
3. Calculate the discriminant:
\( \sqrt{8^2 - 4 \cdot 15} = \sqrt{64 - 60} = \sqrt{4} = 2 \)
4. Solve for \( t \):
\( t = \frac{8 \pm 2}{2} \implies t = 5 \text{ or } 3 \)
Result: The numbers are 5 and 3.