et |x|2 –2|x| + |k – 3| = 0 be a quadratic equation and ‘k’ being a real number. If x is an integer solution of the given quadratic equation, then what is the largest value of x + k?

The given equation is |x|^2 - 2|x| + |k - 3| = 0.

This is a quadratic equation in |x|, so we can solve it by setting it equal to zero and factoring.

|x|^2 - 2|x| + |k - 3| = 0

This can be rewritten as:

(|x| - 1)^2 + |k - 3| - 1 = 0

This equation will have real solutions for |x| if and only if the quantity |k - 3| - 1 is less than or equal to zero, because the square of a real number is always nonnegative.

So, we have:

|k - 3| - 1 ≤ 0

This inequality is satisfied if and only if |k - 3| ≤ 1.

The absolute value of a number is less than or equal to 1 if and only if the number is in the interval [-1, 1]. Therefore, k must be in the interval [2, 4].

Now, we need to find the largest value of x + k. Since k is in the interval [2, 4], the largest possible value of k is 4.

The largest possible value of |x| is 1 (from the equation (|x| - 1)^2 + |k - 3| - 1 = 0), so the largest possible value of x is also 1.

Therefore, the largest possible value of x + k is 1 + 4 = 5.