The roots of the equation `m^2-4m+5=0` are the slopes of the two tangents to the parabola `y^2=4x`. The tangents intersect at the point (h, k). Then h + k is equal to:

Answer

To solve the problem step by step, we need to find the value of \( h + k \) where \( h \) and \( k \) are the coordinates of the intersection point of the tangents to the parabola \( y^2 = 4x \) with slopes given by the roots of the equation \( m^2 - 4m + 5 = 0 \).

### Step 1: Find the roots of the quadratic equation
The given equation is:
\[ m^2 - 4m + 5 = 0 \]
To find the roots, we can use the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -4 \), and \( c = 5 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4 \]
Since the discriminant is negative, the roots are complex:
\[ m = \frac{4 \pm \sqrt{-4}}{2} = \frac{4 \pm 2i}{2} = 2 \pm i \]
Thus, the slopes of the tangents are \( m_1 = 2 + i \) and \( m_2 = 2 - i \).

### Step 2: Find the intersection point of the tangents
For a parabola \( y^2 = 4ax \), the coordinates of the intersection point of the tangents with slopes \( m_1 \) and \( m_2 \) can be given by:
\[ h = a m_1 m_2 \quad \text{and} \quad k = a(m_1 + m_2) \]
Here, \( a = 1 \) (since the parabola is \( y^2 = 4x \)), so we need to calculate \( m_1 m_2 \) and \( m_1 + m_2 \).

### Step 3: Calculate \( m_1 + m_2 \) and \( m_1 m_2 \)
Using the roots:
\[ m_1 + m_2 = (2 + i) + (2 - i) = 4 \]
\[ m_1 m_2 = (2 + i)(2 - i) = 2^2 - i^2 = 4 + 1 = 5 \]

### Step 4: Find \( h \) and \( k \)
Now substituting into the formulas for \( h \) and \( k \):
\[ h = 1 \cdot (m_1 m_2) = 5 \]
\[ k = 1 \cdot (m_1 + m_2) = 4 \]

### Step 5: Calculate \( h + k \)
Now, we can find \( h + k \):
\[ h + k = 5 + 4 = 9 \]

### Final Answer
Thus, the value of \( h + k \) is:
\[ \boxed{9} \]