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We are asked to solve the quadratic equation \( f^2 - 12f + 19 = 0 \) by completing the square.

### Step 1: Move the constant term to the other side
We begin by isolating the constant:
\[
f^2 - 12f = -19
\]

### Step 2: Complete the square
To complete the square, we need to add \( \left( \frac{-12}{2} \right)^2 = 36 \) to both sides:
\[
f^2 - 12f + 36 = -19 + 36
\]
\[
f^2 - 12f + 36 = 17
\]

Now, we can write the left-hand side as a perfect square:
\[
(f - 6)^2 = 17
\]

### Step 3: Solve for \( f \)
Take the square root of both sides:
\[
f - 6 = \pm \sqrt{17}
\]

Now solve for \( f \):
\[
f = 6 \pm \sqrt{17}
\]

### Step 4: Simplify the answers
Since we are asked to express the answers as integers, fractions, or decimals rounded to the nearest hundredth, we compute \( \sqrt{17} \):
\[
\sqrt{17} \approx 4.12
\]

Thus, the two solutions are:
\[
f_1 = 6 + 4.12 = 10.12
\]
\[
f_2 = 6 - 4.12 = 1.88
\]

### Final Answers:
\[
f = 10.12 \quad \text{or} \quad f = 1.88
\]

Do you want further details or have any questions? Here are some related questions:

1. How do you complete the square for different types of quadratic equations?
2. What is the general formula for completing the square?
3. How does completing the square relate to the quadratic formula?
4. Can completing the square be applied to equations with no real solutions?
5. How does the solution change if the quadratic equation has complex roots?

**Tip:** Completing the square is a useful method for deriving the quadratic formula and solving quadratic equations when factoring is difficult.

Solve by completing the square: f^2 - 12f + 19 = 0.

Learn how to solve the quadratic equation f^2 - 12f + 19 = 0 by completing the square. This step-by-step solution covers core algebra concepts, including the quadratic formula and the method of completing the square. Suitable for students in Grades 8-10.

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