Input the geometric mean and arithmetic mean of two numbers to calculate the numbers themselves.
Let the two numbers be \(x\) and \(y\), and their known values are:
Geometric Mean (\(G\)): The square root of the product of the two numbers: \( \text{Geometric Mean} = \sqrt{x \cdot y} \)
Arithmetic Mean (\(A\)): Half the sum of the two numbers: \( \text{Arithmetic Mean} = \frac{x + y}{2} \)
Using the geometric mean, calculate the product of the two numbers: \( x \cdot y = G^2 \) Using the arithmetic mean, calculate the sum of the two numbers: \( x + y = 2A \) Substituting the product and sum into a quadratic equation: \( t^2 - (2A)t + G^2 = 0 \) Use the quadratic formula to find \(t\): \( t = \frac{2A \pm \sqrt{(2A)^2 - 4 \cdot G^2}}{2} \) The two solutions correspond to \(x\) and \(y\).
Solution:
1. Calculate the Product \( x \cdot y \):
\( x \cdot y = 8^2 = 64 \)
2. Calculate the Sum \( x + y \):
\( x + y = 2 \times 17 = 34 \)
3. Quadratic Equation:
\( t^2 - 34t + 64 = 0 \)
4. Solve the Equation:
\( t = \frac{34 \pm \sqrt{34^2 - 4 \cdot 64}}{2} = \frac{34 \pm \sqrt{1156 - 256}}{2} = \frac{34 \pm \sqrt{900}}{2} \)
\( t = \frac{34 \pm 30}{2} \)
\( t_1 = \frac{34 + 30}{2} = 32 \)
\( t_2 = \frac{34 - 30}{2} = 2 \)
Result: The two numbers are 32 and 2.
Solution:
1. Calculate the Product \( x \cdot y \):
\( x \cdot y = 24^2 = 576 \)
2. Calculate the Sum \( x + y \):
\( x + y = 2 \times 30 = 60 \)
3. Quadratic Equation:
\( t^2 - 60t + 576 = 0 \)
4. Solve the Equation:
\( t = \frac{60 \pm \sqrt{60^2 - 4 \cdot 576}}{2} = \frac{60 \pm \sqrt{3600 - 2304}}{2} = \frac{60 \pm \sqrt{1296}}{2} \)
\( t = \frac{60 \pm 36}{2} \)
\( t_1 = \frac{60 + 36}{2} = 48 \)
\( t_2 = \frac{60 - 36}{2} = 12 \)
Result: The two numbers are 48 are 12.