To express the quadratic function $f(x) = (x+3)(x-7)$ in vertex form, we will first expand the expression and then complete the square.

Step 1: Expand the expression

We start by expanding $f(x) = (x+3)(x-7)$ using the distributive property (FOIL method):

$f(x) = x^2 - 7x + 3x - 21$ Simplify the terms: $f(x) = x^2 - 4x - 21$

Step 2: Complete the square

Now, we will convert this expression into vertex form $f(x) = a(x-h)^2 + k$.

1. Start with the quadratic expression: $f(x) = x^2 - 4x - 21$.
2. To complete the square, take the coefficient of $x$, which is $-4$, divide it by 2, and square it: $\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4$
3. Add and subtract 4 inside the expression: $f(x) = (x^2 - 4x + 4) - 4 - 21$ Simplify: $f(x) = (x - 2)^2 - 25$

Final answer:

The vertex form of the function is: $f(x) = (x - 2)^2 - 25$

This means the vertex is $(2, -25)$.

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Would you like more details on any of the steps, or do you have any other questions about this process? Here are some related questions to explore:

1. What is the general process for completing the square for any quadratic function?
2. How can you identify the vertex of a quadratic function from its standard form?
3. How do you convert a quadratic from vertex form to standard form?
4. What is the significance of the vertex in the graph of a parabola?
5. Can you solve for the roots of a quadratic in vertex form?

Tip: When completing the square, always make sure to balance the equation by adding and subtracting the same value.