To determine the value(s) of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( kx(x-3) + 9 = 0 \)[/tex] has equal roots, we can follow these steps:
1. Rewrite the equation in standard quadratic form:
The given equation is [tex]\( kx(x-3) + 9 = 0 \)[/tex]. Let's first expand and rearrange it into the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex].
[tex]\[
kx(x-3) + 9 = 0
\][/tex]
Expanding [tex]\( kx(x-3) \)[/tex]:
[tex]\[
kx^2 - 3kx + 9 = 0
\][/tex]
Thus, the quadratic equation is:
[tex]\[
kx^2 - 3kx + 9 = 0
\][/tex]
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] :
From the equation [tex]\( kx^2 - 3kx + 9 = 0 \)[/tex], we can identify the coefficients:
[tex]\[
a = k, \quad b = -3k, \quad c = 9
\][/tex]
3. Condition for equal roots:
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have equal roots, the discriminant must be zero. The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-3k)^2 - 4(k)(9)
\][/tex]
4. Set the discriminant to zero and solve for [tex]\( k \)[/tex]:
[tex]\[
(-3k)^2 - 4(k)(9) = 0
\][/tex]
Simplify the equation:
[tex]\[
9k^2 - 36k = 0
\][/tex]
Factor out [tex]\( k \)[/tex]:
[tex]\[
9k(k - 4) = 0
\][/tex]
Set each factor to zero and solve for [tex]\( k \)[/tex]:
[tex]\[
9k = 0 \quad \text{or} \quad k - 4 = 0
\][/tex]
Solving these equations gives:
[tex]\[
k = 0 \quad \text{or} \quad k = 4
\][/tex]
Therefore, the values of [tex]\( k \)[/tex] that make the quadratic equation [tex]\( kx(x-3) + 9 = 0 \)[/tex] have equal roots are [tex]\( \boxed{0 \text{ and } 4} \)[/tex].
