Mathematics
Mathematics
Problem
Three consecutive even integers satisfy the following condition: "The sum of the squares of the two smaller integers is 20 more than the square of the largest integer." (a) Represent the three consecutive even integers algebraically. (b) Form the quadratic equation implied by the given condition and solve it. (c) Write the three integers obtained and show that they meet the stated condition.Intuitive explanation
This problem is about translating a word condition into an algebraic equation. The key insight is to pick a single variable—say —to represent the first even integer, then express the other two as and . Once you've set up the relationship between their squares as described, you'll have a quadratic equation that determines which three consecutive even integers work. The algebra will feel natural once you recognize that "the sum of squares of the two smaller equals the square of the largest plus 20" is just a comparison you can write as an equation.
Intuitive explanation
This problem is about translating a word condition into an algebraic equation. The key insight is to pick a single variable—say —to represent the first even integer, then express the other two as and . Once you've set up the relationship between their squares as described, you'll have a quadratic equation that determines which three consecutive even integers work. The algebra will feel natural once you recognize that "the sum of squares of the two smaller equals the square of the largest plus 20" is just a comparison you can write as an equation.