To find the value of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( 9x^2 + 24x + k \)[/tex] has equal roots, we will consider the condition for equal roots in a quadratic equation.
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have equal roots, its discriminant must be zero. The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Given the quadratic equation [tex]\( 9x^2 + 24x + k \)[/tex], we identify the coefficients:
[tex]\[
a = 9, \quad b = 24, \quad c = k
\][/tex]
The condition for equal roots is:
[tex]\[
\Delta = b^2 - 4ac = 0
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the discriminant formula, we get:
[tex]\[
(24)^2 - 4(9)(k) = 0
\][/tex]
Next, we calculate [tex]\( (24)^2 \)[/tex]:
[tex]\[
24^2 = 576
\][/tex]
Substituting this back into our equation, we have:
[tex]\[
576 - 4 \cdot 9 \cdot k = 0
\][/tex]
Simplifying:
[tex]\[
576 - 36k = 0
\][/tex]
To solve for [tex]\( k \)[/tex], we isolate [tex]\( k \)[/tex] on one side of the equation:
[tex]\[
576 = 36k
\][/tex]
Dividing both sides by 36:
[tex]\[
k = \frac{576}{36}
\][/tex]
Calculating the division:
[tex]\[
k = 16
\][/tex]
So, the value of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( 9x^2 + 24x + k \)[/tex] has equal roots is:
\[
\boxed{16}
