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.

Question

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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:

$-10$