An algebra tile configuration. Only the Product spot is shown. 15 tiles are in the Product spot in 6 columns with 6 rows. First row: 1 + x squared, 5 negative x. Below the + x squared tile are 5 negative x tiles. Off to the right are 4 negative tiles in a separate column.

Question

How many zero pairs must be added to the function
f(x) = x2 – 10x – 4 in order to begin writing the function in vertex form?

4
10
21
25

Answer 1

To convert the function \( f(x) = x^2 - 10x - 4 \) into vertex form, we need to complete the square. The vertex form of a quadratic function is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola defined by the quadratic.

1. Start with the original function: \[ f(x) = x^2 - 10x - 4 \]

2. Focus on the quadratic and linear terms: \[ f(x) = (x^2 - 10x) - 4 \]

3. To complete the square, we need the constant term to form a perfect square trinomial. We take the coefficient of \( x \), which is \(-10\), divide it by 2 to get \(-5\), and square it to get \(25\).

4. We can now complete the square: \[ f(x) = (x^2 - 10x + 25) - 25 - 4 \] \[ = (x - 5)^2 - 29 \]

In order to form the perfect square trinomial \(x^2 - 10x + 25\), we had to add 25 inside the parentheses. However, to maintain equality with the original equation, we also need to subtract 25 to counterbalance it, and we have already subtracted 4.

Now, to answer the question:

To convert the function into vertex form, we essentially needed to add zero pairs (i.e., adding the same number and subtracting it) to maintain the equation's equality. Therefore, you needed to add 25 (to complete the square) and then subtract it again.

Thus, the number of zero pairs that must be added is 25. Therefore, the answer is:

25