Input the sum and reciprocal sum of two numbers to quickly calculate the numbers.
Let \( x \) and \( y \) be the two numbers, and assume their sum \( S \) and reciprocal sum \( R \) are known. The initial formulas are as follows: Sum of two numbers: \( x + y = S \) Reciprocal sum: \(\frac{1}{x} + \frac{1}{y} = R\)
Solution:
1. Calculate \( xy \):
\( xy = \frac{S}{R} = \frac{8}{\frac{8}{15}} = 15 \)
2. Form the quadratic equation:
\( t^2 - 8t + 15 = 0 \)
3. Solve for \( x \) and \( y \):
\( t = \frac{8 \pm \sqrt{8^2 - 4 \times 15}}{2} = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm 2}{2} \)
\(x = 5\) and \(y = 3\)
Result: The two numbers are 5 and 3.
Solution:
1. Calculate \( xy \):
\( xy = \frac{S}{R} = \frac{11}{\frac{11}{28}} = 28 \)
2. Form the quadratic equation:
\( t^2 - 11t + 28 = 0 \)
3. Solve for \( x \) and \( y \):
\( t = \frac{11 \pm \sqrt{11^2 - 4 \times 28}}{2} = \frac{11 \pm \sqrt{121 - 112}}{2} = \frac{11 \pm 3}{2} \)
\(x = 7\) and \(y = 4\)
Result: The two numbers are 7 and 4.
Solution:
1. Calculate \( xy \):
\( xy = \frac{S}{R} = \frac{15}{\frac{3}{10}} = 50 \)
2. Form the quadratic equation:
\( t^2 - 15t + 50 = 0 \)
3. Solve for \( x \) and \( y \):
\( t = \frac{15 \pm \sqrt{15^2 - 4 \times 50}}{2} = \frac{15 \pm \sqrt{225 - 200}}{2} = \frac{15 \pm 5}{2} \)
\(x = 10\) and \(y = 5\)
Result: The two numbers are 10 and 5.