Substitute the given values $ a + b = 7 $, $ ab = 10 $:

February 22, 2026

Substitute the given values $ a + b = 7 $, $ ab = 10 $:

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y = 3(0) + 1 = 1
Thus, the $ y $-intercept point is $ oxed{(0, 1)} $.
Question: If $ a + b = 7 $ and $ ab = 10 $, what is the value of $ a^3 + b^3 $?
a^3 + b^3 = 7^3 - 3(10)(7) = 343 - 210 = 133
Thus, the value is $ oxed{133} $.Question: How many lattice points lie on the hyperbola $ x^2 - y^2 = 2025 $?
Solution: The equation $ x^2 - y^2 = 2025 $ factors as $ (x - y)(x + y) = 2025 $. Since $ x $ and $ y $ are integers, both $ x - y $ and $ x + y $ must be integers. Let $ a = x - y $ and $ b = x + y $, so $ ab = 2025 $. Then $ x = rac{a + b}{2} $ and $ y = rac{b - a}{2} $. For $ x $ and $ y $ to be integers, $ a + b $ and $ b - a $ must both be even, meaning $ a $ and $ b $ must have the same parity. Since $ 2025 = 3^4 \cdot 5^2 $, it has $ (4+1)(2+1) = 15 $ positive divisors. Each pair $ (a,

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