problem_type: Algebra

problem: Find the roots of the quadratic equation \(x^2 - 4x + 4 = 0\).

solution: The equation simplifies to \((x - 2)^2 = 0\), so the root is \(x = 2\).


problem: Determine the value of \(y\) in the equation \(2y + 3 = 7\).

solution: Subtracting 3 from both sides gives \(2y = 4\).


solution: Dividing both sides by 2 gives \(y = 2\).


problem_type: Combinatorics

problem: How many ways can 3 objects be arranged in a line?

solution: The number of permutations of 3 objects is \(3! = 6\).