To solve for the value of [tex]\( k \)[/tex] in the quadratic equation [tex]\( x^2 - 4x + k = 0 \)[/tex] given that the equation has equal roots, we can follow these steps:
1. Understanding the Condition for Equal Roots:
A quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] has equal roots if and only if its discriminant ([tex]\( \Delta \)[/tex]) is zero. The discriminant of a quadratic equation is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
2. Identify the Coefficients:
Comparing the given quadratic equation [tex]\( x^2 - 4x + k = 0 \)[/tex] with the general form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -4 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = k \)[/tex] (constant term)
3. Set the Discriminant to Zero:
For the quadratic equation to have equal roots, the discriminant must be zero:
[tex]\[
\Delta = b^2 - 4ac = 0
\][/tex]
4. Substitute the Values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the Discriminant Formula:
[tex]\[
\Delta = (-4)^2 - 4(1)(k) = 0
\][/tex]
Simplify this equation:
[tex]\[
16 - 4k = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
Isolate [tex]\( k \)[/tex] by solving the equation:
[tex]\[
16 - 4k = 0
\][/tex]
Add [tex]\( 4k \)[/tex] to both sides:
[tex]\[
16 = 4k
\][/tex]
Divide both sides by 4:
[tex]\[
k = \frac{16}{4}
\][/tex]
[tex]\[
k = 4
\][/tex]
So, the value of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( x^2 - 4x + k = 0 \)[/tex] has equal roots is [tex]\( \boxed{4} \)[/tex].
