Input the difference and difference of cubes of two numbers to instantly determine their values.
Assume the two numbers are \( x \) and \( y \), with the following given:
Step-by-Step Process:
Solution:
1. Calculate \( xy \):
\( xy = \frac{335 - 5^3}{3 \cdot 5} = 14 \)
2. Set up the quadratic equation:
\( y^2 + 5y - 14 = 0 \)
3. Calculate the discriminant:
\( \sqrt{5^2 + 4 \cdot 14} = \sqrt{25 + 56} = \sqrt{81} = 9 \)
4. Solve for \( y \):
\( y_1 = \frac{-5 + 9}{2} = 2 \)
\( y_2 = \frac{-5 - 9}{2} = -7 \)
5. Calculate \( x \):
\( x_1 = 2 + 5 = 7 \)
\( x_2 = -7 + 5 = -2 \)
Result: The numbers are (7, 2) or (-2, -7).
Solution:
1. Calculate \( xy \):
\( xy = \frac{316 - 4^3}{3 \cdot 4} = 21 \)
2. Set up the quadratic equation:
\( y^2 + 4y - 21 = 0 \)
3. Calculate the discriminant:
\( \sqrt{4^2 + 4 \cdot 21} = \sqrt{16 + 84} = \sqrt{100} = 10 \)
4. Solve for \( y \):
\( y_1 = \frac{-4 + 10}{2} = 3 \)
\( y_2 = \frac{-4 - 10}{2} = -7 \)
5. Calculate \( x \):
\( x_1 = 3 + 4 = 7 \)
\( x_2 = -7 + 4 = -3 \)
Result: The numbers are (7, 3) or (-3, -7).
Solution:
1. Calculate \( xy \):
\( xy = \frac{999 - 3^3}{3 \cdot 3} = 108 \)
2. Set up the quadratic equation:
\( y^2 + 3y - 108 = 0 \)
3. Calculate the discriminant:
\( \sqrt{3^2 + 4 \cdot 108} = \sqrt{9 + 432} = \sqrt{441} = 21 \)
4. Solve for \( y \):
\( y_1 = \frac{-3 + 21}{2} = 9 \)
\( y_2 = \frac{-3 - 21}{2} = -12 \)
5. Calculate \( x \):
\( x_1 = 9 + 3 = 12 \)
\( x_2 = -12 + 3 = -9 \)
Result: The numbers are (12, 9) or (-9, -12).