To determine the value of k for which the expression 3x^2 - 2x + k is a perfect square, you need to complete the square. A perfect square expression can be written in the form (ax + b)^2.
Start by completing the square for the given expression:
3x^2 - 2x + k
First, factor out the coefficient of x^2, which is 3:
3(x^2 - (2/3)x) + k
Now, to complete the square, you'll need to add and subtract the square of half the coefficient of x, which is (-2/3)^2 = 4/9:
3(x^2 - (2/3)x + 4/9 - 4/9) + k
Now, rewrite the expression:
3[(x - 2/3)^2 - 4/9] + k
Expand the square:
3(x - 2/3)^2 - 4/3 + k
Now, for the expression to be a perfect square, the constant term (-4/3 + k) must be 0 because perfect squares have no constant term.
So, set -4/3 + k = 0 and solve for k:
k = 4/3
Therefore, the value of k for which the expression 3x^2 - 2x + k is a perfect square is k = 4/3.