Réponse:
To determine the value of \( k \) such that the expression \( x^2 + kx + \frac{16}{9} \) is a perfect square, we need to complete the square for this quadratic expression.
Given the expression \( x^2 + kx + \frac{16}{9} \), we complete the square by adding and subtracting the square of half the coefficient of \( x \):
\[
x^2 + kx + \frac{16}{9} = \left(x + \frac{k}{2}\right)^2 - \left(\frac{k}{2}\right)^2 + \frac{16}{9}
\]
For this expression to be a perfect square, the constant term \( -\left(\frac{k}{2}\right)^2 + \frac{16}{9} \) must be equal to zero. So:
\[
-\left(\frac{k}{2}\right)^2 + \frac{16}{9} = 0
\]
Solving this equation:
\[
\frac{k^2}{4} = \frac{16}{9}
\]
Multiply both sides by \( 4 \):
\[
k^2 = \frac{64}{9}
\]
Taking the square root of both sides:
\[
k = \pm \frac{8}{3}
\]
Therefore, \( k \) can be either \( \frac{8}{3} \) or \( -\frac{8}{3} \) for the expression \( x^2 + kx + \frac{16}{9} \) to be a perfect square.
