Question

Complete the square to find the minimum value of f of x equals x squared plus 4 x minus 6. Enter the answer in the box.

Response area with 1 text input box
The minimum value of f of x is

Answer 1

To complete the square for the function \( f(x) = x^2 + 4x - 6 \), we can follow these steps:

1. Start with the quadratic part of the function, which is \( x^2 + 4x \).
2. To complete the square, we need to find a number that, when added and subtracted to the expression, forms a perfect square trinomial. The coefficient of \( x \) is 4. Half of 4 is 2, and squaring it gives us \( 2^2 = 4 \).
3. Therefore, we rewrite the function by adding and subtracting 4:

\[
f(x) = (x^2 + 4x + 4) - 4 - 6
\]

This simplifies to:

\[
f(x) = (x + 2)^2 - 10
\]

4. The expression \( (x + 2)^2 \) is always non-negative and reaches its minimum value of 0 when \( x = -2 \).
5. Thus, the minimum value of \( f(x) \) occurs at \( x = -2 \):

\[
f(-2) = 0 - 10 = -10
\]

Therefore, the minimum value of \( f(x) \) is \(-10\).

So the final answer to enter in the box is:

\(-10\)