To find the value of \( k \) such that the equation \( x^2 + kx + k - 1 = 0 \) has one real root, we can use the discriminant of the quadratic equation.
The discriminant, denoted by \( \Delta \), is given by the formula:
\[ \Delta = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).
In our equation, \( a = 1 \), \( b = k \), and \( c = k - 1 \).
So, substituting these values into the discriminant formula:
\[ \Delta = k^2 - 4(1)(k - 1) \]
\[ \Delta = k^2 - 4k + 4 \]
For the equation to have one real root, the discriminant must be equal to zero. Therefore:
\[ k^2 - 4k + 4 = 0 \]
This is a quadratic equation in \( k \). We can solve it by factoring or using the quadratic formula. Since this equation factors nicely, let's factor it:
\[ (k - 2)^2 = 0 \]
Now, take the square root of both sides:
\[ k - 2 = 0 \]
\[ k = 2 \]
So, the value of \( k \) that makes the equation \( x^2 + kx + k - 1 = 0 \) have one real root is \( k = 2 \).