To determine the values of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( x^2 - 4x + k =0 \)[/tex] has a double root, we need to analyze the condition for a double root.
For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] to have a double root, the discriminant must be zero. The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
In the given equation [tex]\( x^2 - 4x + k = 0 \)[/tex], we identify the coefficients as:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = k \)[/tex]
Next, we substitute these values into the discriminant formula:
[tex]\[
\Delta = (-4)^2 - 4 \cdot 1 \cdot k
\][/tex]
Simplifying the expression, we get:
[tex]\[
\Delta = 16 - 4k
\][/tex]
For the equation to have a double root, the discriminant must be zero:
[tex]\[
16 - 4k = 0
\][/tex]
We solve for [tex]\( k \)[/tex]:
[tex]\[
16 = 4k
\][/tex]
[tex]\[
k = \frac{16}{4}
\][/tex]
[tex]\[
k = 4
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] for which the equation [tex]\( x^2 - 4x + k = 0 \)[/tex] has a double root is [tex]\( k = 4 \)[/tex].
Finally, the discriminant and the value of [tex]\( k \)[/tex] can be summarized as:
[tex]\[
\Delta = 16 - 4k
\][/tex]
[tex]\[
k = 4
\][/tex]
