f(x)=x^2+4kx+(3+11k)
solve by completing the square and show me the steps

To solve the quadratic function \( f(x) = x^2 + 4kx + (3 + 11k) \) by completing the square, follow these steps:

### Step 1: Write the quadratic equation in a standard form.
The equation is already in the form of \( f(x) = x^2 + bx + c \), where:
- \( b = 4k \)
- \( c = 3 + 11k \)

### Step 2: Isolate the constant term \( c \) from the quadratic and linear terms.
We begin by focusing on the first two terms involving \( x \), so rewrite the function as:
\[
f(x) = x^2 + 4kx + (3 + 11k)
\]

We'll work with the quadratic and linear terms first:
\[
f(x) = (x^2 + 4kx) + (3 + 11k)
\]

### Step 3: Complete the square on the quadratic and linear terms.
To complete the square, we need to add and subtract a constant that makes the expression \( x^2 + 4kx \) a perfect square trinomial. The constant needed is \( \left(\frac{b}{2}\right)^2 \), where \( b = 4k \). So:

\[
\frac{b}{2} = \frac{4k}{2} = 2k
\]

Now square this value:
\[
(2k)^2 = 4k^2
\]

We add and subtract \( 4k^2 \) inside the equation:
\[
f(x) = (x^2 + 4kx + 4k^2 - 4k^2) + (3 + 11k)
\]

### Step 4: Simplify the expression into a perfect square form.
The terms inside the parentheses \( x^2 + 4kx + 4k^2 \) form a perfect square trinomial, which can be written as:
\[
f(x) = (x + 2k)^2 - 4k^2 + (3 + 11k)
\]

### Step 5: Combine the constant terms.
Now combine the constants \( -4k^2 \) and \( 3 + 11k \):
\[
f(x) = (x + 2k)^2 + (-4k^2 + 3 + 11k)
\]

Thus, the completed square form of the quadratic function is:
\[
f(x) = (x + 2k)^2 + (3 + 11k - 4k^2)
\]

### Final form:
\[
f(x) = (x + 2k)^2 + (11k + 3 - 4k^2)
\]

This is the solution in the form of completing the square.

---

Would you like more details or have any questions about the steps? Here are some questions to further explore the concept:

1. How do you recognize when a quadratic can be solved by completing the square?
2. What is the significance of the expression \( \left(\frac{b}{2}\right)^2 \) in completing the square?
3. Can you factor the quadratic directly instead of completing the square?
4. How does completing the square help in finding the vertex of a parabola?
5. What are other methods to solve a quadratic equation?

**Tip:** Completing the square is especially useful for converting a quadratic equation into vertex form, which is helpful in graphing parabolas.

f(x) = x^2 + 4kx + (3 + 11k) solve by completing the square and show me the steps

Step-by-step solution for solving the quadratic function f(x) = x^2 + 4kx + (3 + 11k) by completing the square. Learn key algebraic techniques, including how to transform a quadratic equation into vertex form. Suitable for students in Grades 9-11.

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