AMC 8 Daily Practice - Multiplication Formula

Complete problem set with solutions and individual problem pages

Problem 2 Easy

Given a real number x satisfies the equation:   (x - 2023)^2 + (2024 - x)^2 = 2025, what is the value of (x - 2023)(2024 - x)?

  • A.

    -1013

  • B.

    -1012

  • C.

    1012

  • D.

    1013

  • E.

    2025

Answer:B

Let a = x - 2023.

Then, we can express 2024 - x as:   2024 - x =1 - a

Substituting these into the original equation:

a^2 + (1 - a)^2 = 2025

a^2 + \left(1 - 2a + a^2\right) = 2025

2a^2 - 2a + 1 = 2025

2a^2 - 2a - 2024 = 0

a^2 - a = 1012

Observe that: (x - 2023)(2024 - x)=a(1-a)= a - a^2

From the simplified quadratic equation a^2 - a = 1012, multiply both sides by -1:   -a^2 + a = -1012.

Thus, the value of (x - 2023)(2024 - x) is:   -1012

Final result: \boxed{-1012 }

Related Problems
Problem 1 AMC 8 Daily Practice - Multiplication Formula
Problem 3 AMC 8 Daily Practice - Multiplication Formula
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Problem 5 AMC 8 Daily Practice - Multiplication Formula
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