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To solve this problem, we need to analyze the quadratic function \( f(x) = x^2 - 14x + 53 \) and find the values of \( a \) and \( b \) such that the roots of the equation are \( x = a \pm bi \), where \( i \) represents the imaginary unit.

1. **Rewrite as an Equation**:
   Since \( f(x) = 0 \), we have:
   \[
   x^2 - 14x + 53 = 0
   \]

2. **Identify Coefficients**:
   For this quadratic equation \( ax^2 + bx + c = 0 \), we identify:
   - \( a = 1 \)
   - \( b = -14 \)
   - \( c = 53 \)

3. **Calculate the Discriminant**:
   The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is \( D = b^2 - 4ac \).
   Substituting the values:
   \[
   D = (-14)^2 - 4 \cdot 1 \cdot 53 = 196 - 212 = -16
   \]
   Since the discriminant is negative, this confirms the roots are complex.

4. **Find the Roots Using the Quadratic Formula**:
   The roots of a quadratic equation are given by:
   \[
   x = \frac{-b \pm \sqrt{D}}{2a}
   \]
   Substituting \( b = -14 \), \( D = -16 \), and \( a = 1 \):
   \[
   x = \frac{14 \pm \sqrt{-16}}{2}
   \]
   \[
   x = \frac{14 \pm 4i}{2}
   \]
   \[
   x = 7 \pm 2i
   \]

5. **Interpret the Results**:
   The roots are \( x = 7 \pm 2i \), which implies:
   - \( a = 7 \)
   - \( b = 2 \)

### Final Answer:
- **The value of \( a \) is** \( 7 \).
- **The value of \( b \) is** \( 2 \).

Would you like further details, or have any questions?

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### Related Questions:
1. How do you find the real and imaginary parts of a complex root?
2. What is the significance of the discriminant in determining the nature of roots?
3. How would the roots change if the discriminant were positive?
4. What steps are involved in completing the square for a quadratic equation?
5. How can complex conjugate roots be represented in polar form?

### Tip:
In any quadratic equation, a negative discriminant always results in complex conjugate roots.

The quadratic function f(x) = x^2 - 14x + 53 is equal to zero when x = a ± bi. What is the value of a? What is the value of b?

Learn how to solve the quadratic equation f(x) = x^2 - 14x + 53 with complex roots. This example demonstrates finding the values of a and b when the roots are complex, covering the quadratic formula and complex number concepts. Suitable for students in Grades 9-10.

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